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University  of  California. 

'      GrII''irr    OP1 


1876. 


A   TREATISE 


ON  THE 


RESISTANCE  OF  MATERIALS, 

5lntr  an 


ON   THE 


PRESERVATION    OF  TIMBER. 


BY 

DE'VOLSON  WOOD, 

PROFESSOR  OF  CIVIL  ENGINEERING  IN  THE  UNIVERSITY  OF  MICHIGAN. 

tftVER* 


NEW    YORK: 

JOHN   WILEY    &    SON, 


15  ASTOR  PLACE. 


Entered  according  to  Act  of  Congress,  in  the  year  1871, 

By  JOHN  WILEY  &  SON,  ' 
In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


THE  NEW  YORK  PRINTING  COMPANY, 

205,  207,  209,  211,  and  213  East  Twelfth  Street. 


PREFACE. 


THIS  book  contains  the  substance  of  my  lectures  to  the  Senior  Class 
in  Civil  Engineering,  in  the  University,  during  the  past  few  years,  on 
the  Resistance  of  Materials.  The  chief  aim  has  been  to  present  the 
theories  as  they  exist  at  the  present  time.  The  subject  is  necessarily 
an  experimental  one,  and  any  theory  which  has  not  the  results  of  ex- 
periments for  its  foundation  is  valueless.  I  have  therefore  presented 
the  results  of  a  few  experiments  under  each  head,  as  they  have  been 
obtained  in  various  parts  of  the  world,  that  the  student  may  judge 
for  himself  whether  the  theory  is  well  founded  or  not.  It  is  hoped 
that  this  part  of  the  work  will  be  valuable  to  the  practical  man. 

The  descriptive  parts  are  given  more  fully  here  than  they  were  in 
the  lectures,  because  they  can  be  consulted  more  profitably  on  the 
printed  page  than  they  could  in  the  manuscript,  and  will  be  examined 
more  by  the  general  reader  than  the  mathematical  part.  But,  on  the 
other  hand,  the  mathematical  part  is  much  more  condensed  here  than 
it  was  in  the  class-room.  This  was  done  so  as  to  keep  the  work  in  as 
small  a  space  as  possible ;  and  also  because  a  student  is  supposed  to 
have  time  for  deliberate  study,  and  can  take  time  to  overcome  his  dif- 
ficulties and  secure  his  results.  It  is  intended,  however,  in  the  next 
edition,  to  publish  an  appendix,  in  order  to  explain  the  more  difficult 
mathematical  operations  of  the  text. 

I  have  taken  special  pains  to  make  frequent  references  to  other 
books  and  reports  from  which  I  have  secured  information.  This  will 
enable  any  one  to  verify  more  fully  the  positions  which  have  been 
taken,  and  will  be  convenient  for  those  who  desire  to  secure  a  more 
thorough  knowledge  of  any  particular  topic. 

I  do  not  deem  it  necessary  to  indicate  those  topics  which  are  wholly 
original.  To  the  reader  who  has  never  before  given  the  subject  any 
attention,  all  will  be  new ;  and  the  well-informed  reader  will  readily 
detect  what  is  original. 

A  large  amount  of  labor  and  study  has  been  given  to  this  subject 
in  nearly  all  civilized  countries,  and  yet  the  theories  in  regard  to  re- 
sistance from  transverse  stress  are  not  very  satisfactory.  In  regard 
to  the  strength  of  rectangular  beams,  the  "  Common  Theory,"  as  I 


IV  PREFACE. 

have  called  it,  is  sufficiently  correct  for  ordinary  practical  purposes, 
especially  if  the  modulus  of  rupture,  as  determined  by  direct  experi- 
ment upon  rectangular  beams  of  the  same  material,  be  used.  Bar- 
low's (t Theory  of  Flexure"  appears  to  be  more  nearly  correct  in 
theory  when  applied  to  rectangular  beams  and  beams  of  the  I  section, 
or  other  forms  which  are  symmetrical  in  reference  to  the  neutral  axis. 
But  when  the  sections  are  irregular  none  of  the  theories  can  be  relied 
upon  for  securing  correct  results.  Whatever  theory  may  yet  do  for  us, 
it  is  quite  evident  that  no  theory  will  ever  be  devised,  of  practical 
value,  which  will  be  applicable  to  the  infinite  variety  of  forms  of 
beams  which  are  or  may  be  used  in  the  mechanic  arts.  That  I  may 
not  be  misunderstood  upon  this  point,  I  will  be  more  specific.  We 
know  that  our  present  theories  do  not  always  give  correct  results,  and 
that  the  more  irregular  the  form  the  greater  the  discrepancy  between 
the  actual  and  computed  strengths  of  a  beam.  Now,  if  a  theory  is 
ever  devised  which  will  take  into  account  all  the  conditions  of  strains 
in  a  beam,  I  think  it  will  be  too  complicated  to  be  of  practical  value 
to  the  mechanic.  I  do  not  desire  by  this  remark  to  disparage  theory. 
Theories  are  valuable.  Without  them  we  would  make  little  or  no 
progress.  Fortunately  for  the  engineer,  it  is  not  the  mathematically 
exact  result  that  he  desires,  but  the  reliable  result.  He  does  not  so 
much  desire  to  know  that  one  pound  more  of  load  will  break  his 
structure,  as  he  does  that  he  may  depend  upon  it  to  carry  from  four 
to  six  times  the  load  which  he  intends  to  put  upon  it.  The  theories,  as 
now  developed,  are  safe  guides  to  the  mechanic  and  engineer ;  still  we 
learn  to  depend  more  and  more  upon  direct  experiment.  The  theory 
also  in  regard  to  the  deflection  of  beams  under  a  transverse  strain, 
has  recently  received  a  modification,  due  to  a  consideration  of  the 
effect  of  transverse  shearing ;  but  the  modification  is  sustained  both 
from  mathematical  and  experimental  considerations.  May  not  more 
careful  experiments  yet  teach  us  that  it  must  be  still  further  modified 
on  account  of  the  longitudinal  shearing  strain? 

The  author  will  be  pleased  to  receive  the  results  of  experiments 
which,  have  been  made  in  this  country,  so  that  if  this  work  is  revised 
in  the  future,  it  may  be  made  more  profitable  to  the  engineering 
profession. 

ANN  ARBOR,  MICH.,  Sept.,  1871. 


TABLE  OF  CONTENTS. 


INTRODUCTION. 

NO.    OP  THE 

AETICLE.  PAGK 

1.  General  Problems 1 

2.  Definition  of  Certain  Terms 2 

3.  Stresses  produce  Elastic  and  Ultimate  Resistances' 3 

4.  General  Principles  of  Elastic  Resistance 3 

5.  Coefficient  of  Elasticity 4 

6.  Proofs  of  the  Laws  of  Elastic  Resistance 5 


CHAPTER  I. 

TENSION. 

7.  Experiments  on  the  Elongation  of  Wrought  Iron 7 

8.  Graphical  Representation  of  Results 8 

9.  Elongation  of  Cast  Iron 10 

10.  Graphical  Representation 12 

11.  Tables  of  Experiments  on  Cast  Iron 12 

12.  Coefficient  of  Elasticity  of  Malleable  Iron 14 

13.  Elasticity  of  Wood  Longitudinally 15 

14.  Elasticity  of  Wood  Radially 16 

15.  Remark  on  the  Coefficient  of  Elasticity 17 

1C.  Elongation  of  a  Prismatic  Bar  by  a  Weight 17 

17.  Elongation  of  a  Prismatic  Bar  when  the  weight  of  the  bar  is  con- 

sidered   10 

18.  Work  of  Elongation 20 

19.  Vertical  Oscillations 21 

20.  Viscocity  of  Solids  22 

21.  Modulus  of  Strength 23 

22.  Strength  of  a  Prismatic  Bar 24 

23.  Strength  of  a  Prismatic  Bar  when  its  weight  is  considered 25 

24.  Bar  of  Uniform  Strength 2§ 

25.  Strength  of  a  Closed  Cylinder 26 

26.  Strength  of  Glass  Globes '. 80 

27.  Experiments  on  Riveted  Plates 81 

28.  Strength  of  Rolled  Sheets  of  Iron  in  different  directions 83 

29.  Strength  of  Wrought  Iron  at  various  temperatures : .  36 

30.  Effect  of  Severe  Strains  on  the  Tenacity  of  Iron 40 

31.  Effect  of  Repeated  Rupture   41 


VI  CONTENTS. 

NO.  OP  THE 

ABTICLE.  PAGE 

32.  Strength  of  Annealed  Iron 41 

33.  Strength  of  Metals  Modified  by  Treatment 42 

34.  Effect  of  Prolonged  Fusion  on  Cast  Iron 43 

35.  Remelting  Cast  Iron — On  Strength  of 43 

36.  Cooling  Cast  Iron— Affects  Strength  of 44 

37.  Strength  Modified  by  Various  Circumstances 44 

38.  Safe  Limit  of  Loading 44 


CHAPTER    II. 

COMPRESSION. 

39.  Elastic  and  Ultimate  Resistance 46 

40.  Compression  of  Cast  Iron 47 

41.  Compression  of  Wrought  Iron 48 

42.  Graphical  Representation  of  Results 49 

43.  Comparative  Compression  of  Cast  and  Wrought  Iron 49 

44.  Compression  of  other  Metals 50 

45.  Modulus  for  Crushing 50 

46.  Modulus  of  Strain : 51 

47.  Resistance  of  Cast  Iron  to  Crushing 52 

48.  Resistance  of  Wrought  Iron  to  Crushing 53 

49.  Resistance  of  Wood  to  Crushing 53 

50.  Resistance  of  Cast  Steel  to  Crushing 54 

51.  Resistance  of  Glass  to  Crushing 54 

52.  Strength  of  Pillars 55 

53.  Formulas  for  the  Weight  of  Pillars 58 

54.  Irregularities  in  the  Thickness  of  Cast  Pillars 60 

55.  Experiments  of  the  N.  Y.  C.  R.  R.  Co.  on  Angle  Irons 60 

56.  Buckling  of  Tubes 66 

57.  Collapse  of  Tubes . .  66 

58.  Results  of  Experiments  on  Collapsing 69 

59.  Law  of  Thickness  to  resist  Collapsing 69 

60.  Formula  for  Thickness  to  resist  Collapsing 72 

61.  Resistance  of  Elliptical  Tubes  to  Collapsing 72 

62.  Strength  of  very  long  Tubes 73 

63.  Strength  from  External  and  Internal  Pressure  Compared 73 

64.  Resistance  of  Glass  Globes  to  Collapsing 73 


CHAPTER    III. 

THEORIES  OP  FLEXURE  AND  RUPTURE  FROM  TRANSVERSE   STRESS. 

65.  Remark  upon  the  Subject 75 

66.  Galileo's  Theory 75 

67.  Robert  Hooke's  Theory 76 

68.  Marriotte's  and  Leibnitz's  Theory 76 

69.  James  Bernoulli's  Theory 76 

70.  Parent's  Theory 77 


CONTENTS.  Vll 

NO.  OF  THE 

ARTICLE.  PAGE 

71.  Coulomb's  Theory 77 

72.  Young's  "  Modulus  of  Elasticity  " 77 

73.  Navier's  Developments  of  Theory 78 

74.  The  Common  Theory 78 

75.  Barlow's  Theory 79 

76.  Remarks  upon  the  Theories 82 

77.  Position  of  the  Neutral  Axis  found  Experimentally 83 

78.  Position  of  the  Neutral  Axis  found  Analytically 84 


CHAPTER  IV. 

SHEARING     STRESS. 

79.  Examples  of  Shearing  Stress 89 

80.  Modulus  of  Shearing  Stress 89 

81.  A  Problem  of  a  Tie  Beam 91 

82.  A  Problem  of  Riveted  Plates 92 

83.  Lpngitudinal  Shearing  in  a  Bent  Beam 92 

84.  Transverse  Shearing  in  a  Bent  Beam 93 

85.  Shearing  Resistance  to  Torsion 95 


CHAPTER  V. 

FLEXURE. 

86.  General  Equation  of  the  "  Elastic  Curve  " 96 

87.  Moment  of  Inertia  of  a  Rectangle  and  of  a  Circle 98 

88.  GENERAL  STATEMENT  OF  THE  PROBLEMS 99 

89.  Beams  Fixed  at  one  end  Load  at  the  Free  End 99 

90.  Beams  Fixed  at  one  end  and  Loaded  Uniformly 101 

91.  Previous  Cases  Combined 102 

92.  Beams  Supported  at  their  Ends  and  Loaded  at  any  Point 102 

93.  Beams  Supported  at  their  Ends  and  Loaded  Uniformly 105 

94.  The  two  preceding  Cases  Combined 105 

95.  Examples 106 

96.  Deflection  according  to  Barlow's  Theory 108 

97.  Beams  Fixed  at  one  end,  Supported  at  the  other,  and  Loaded  at  any 

Point : 108 

98.  Beams  Fixed  at  one  end,  Supported  at  the  other,  and  Loaded  Uni- 

formly   113 

99.  Beams  Fixed  at  both  ends  and  Loaded  at  the  Middle 115 

100.  Beams  Fixed  at  both  ends  and  Loaded  Uniformly 116 

101.  Table  of  Results 118 

102.  Remarks  upon  the  Results 118 

103.  Modification  of  the  Formulas  for  Deflection  to  include  shearing  re- 

sistance   119 

104.  Unsolved  Problems 121 

105.  Deflection  of  Beams  having  Variable  Sections 122 

106.  Beams  subjected  to  Oblique  Strains 124 


CONTENTS. 

NO.  OF  THE 

AKTICLE.  PAGE 

107.  Flexure  of  Columns 126 

108.  Definition  of  "  Graphical  Methods  " 129 

109.  General  Expression  for  the  Deflection  of  Beams  found  by  the  Graphi- 

cal Method 129 

110.  Solution  of  Case  I.  by  the  Graphical  Method 131 

111.  Solution  of  Case  II.  by  the  Graphical  Method. 133 

112.  Solution  of  Case  III.  by  the  Graphical  Method 134 

113.  Solution  of  Case  IV.  by  the  Graphical  Method 134 

114.  Remark  in  regard  to  other  Cases. 135 

115.  Moment  of  Inertia  of  a  Rectangle  by  the  Graphical  Method 136 

116.  Moment  of  Inertia  of  a  Triangle  by  the  Graphical  Method 136 

117.  Moment  of  Inertia  of  a  Circle  by  the  Graphical  Method 137 

118.  Remark  in  regard  to  the  Moment  of  Inertia  of  other  Surfaces 139 


CHAPTER  VI. 

TRANSVERSE    STRENGTH. 

119.  Strength  of  Solid  Rectangular  Beams 140 

120.  Definition  of  the  Modulus  of  Rupture 142 

121.  Practical  Formulas 144 

122.  Relative  Strength  of  a  Beam 145 

123.  Examples 145 

124.  Relation  between  Strain  and  Deflection 146 

125.  Strength  of  Hollow  Rectangular  Beams 147 

126.  Strength  of  Double  T  Beams 148 

127.  True  Value  of  d. 150 

128.  Experiments  of  Baron  Von  Beber  for  determining  the  Thickness  of 

the  Vertical  Web • 151 

129.  ANOTHER  GRAPHICAL  METHOD 154 

130.  Strength  of  a  Square  Beam  with  its  Diagonal  Vertical 154 

131.  Treatment  of  Irregular  Sections 156 

132.  Formula  of  Strength  according  to  Barlow's  Theory 158 

133.  Strength  of  a  Beam  Loaded  at  any  number  of  Points 159 

134.  Strength  of  a  Beam  when  Loaded  Uniformly  over  a  portion  of  its 

length 160 

135.  Example  of  Oblique  Strain 162 

136.  GENERAL  FORMULA  OP  STRENGTH 163 

137.  Strength  of  a  Rectangular  Beam  determined  from  the  General  For- 

mula   164 

138.  Rectangular  Beam,  with  its  Sides  Inclined 164 

139.  Strongest  Rectangular  Beam  which  can  be  cut  from  a  given  Cylin- 

drical one 166 

140.  Strength  of  Triangular  Beams 166 

141.  Strongest  Trapezoidal  Beam  which  can  be  cut  from  a  Triangular  one.  168 

142.  Moment  of  Resistance  of  Cylindrical  Beams 169 

143.  Moment  of  Resistance  of  Elliptical  Beams 171 

144.  Moment  of  Resistance  of  Parabolic  Beams 171 

145.  Strength  according  to  Barlow's  Theory 172 


CONTENTS.  IX 

CHAPTER  VII. 

BEAMS  OP  UNIFORM  RESISTANCE. 

NO.  OF  THE 

ARTICLE.  PAGE 

146.  General  Expression 173 

147.  Beams  Fixed  at  one  end  and  Loaded  at  the  Free  End 173 

148.  Beams  Fixed  at  one  end  and  Uniformly  Loaded 174 

149.  The  two  preceding  Cases  Combined 175 

150.  Beams  Fixed  at  one  end,  and  the  weight  of  the  Beam  the  only  Load.  176 

151.  Beams  Supported  at  their  Ends 178 

152.  Beams  Fixed  at  their  Ends 179 

153.  Effect  of  Transverse  Shearing  Stress  on  Modifying  the  form  of  the 

Beams  of  Uniform  Resistance,  and  Tabulated  Values  of  Shearing 
Stresses. 180 

154.  Unsolved  Problems  of  Beams  of  Uniform  Resistance 183 

155.  Best  form  of  Cast-Iron  Beam,  as  found  Experimentally 184 

156.  Hodgkinson's  Formula  for  the  Strength  of  Beams  of  the    "Type 

Form  " 187 

157.  Experiments  on  T  Rails '. 187 

158.  Remark  on  Rolled  Wrought  Iron  Beams 188 


CHAPTER  VIII. 
TORSION. 

159.  How  Torsive  Strains  are  Produced 189 

160.  Angle  of  Torsion 189 

161.  Value  of  the  Coefficient  of  Elastic  Resistance  to  Torsion 191 

162.  Torsion  Pendulum 192 

163.  Rupture  by  Torsion 193 

164.  Practical  Formulas ." 194 

165.  Results  of  Wertheim's  Experiments 196 


CHAPTER  IX. 

EFFECTS  OF   LONG-CONTINUED   STRAINS  AND  OF  SHOCKS — CRYSTALLIZATION. 

166.  General  Remark 198 

167.  Hodgkinson's  Experiments 199 

168.  Vicat's  Experiments 199 

169.  Fairbairn's  Experiments 199 

170.  Roebling's  Observations 202 

OFT  REPEATED  STRAINS 203 

171.  Effect  of  Shocks 206 

172.  Crystallization 210 


CONTENTS. 


CHAPTER  X. 

LIMITS  OP   SAFE  LOADING  FOB,   MECHANICAL   STRUCTURES. 
NO.  OP  THE 

ABTICLE.  PAGE  . 

173.  Bisk  and  Safety. 219 

174.  Absolute  Modulus  of  Safety 219 

175.  Factor  of  Safety 220 

176.  Safe  Load  as  determined  by  the  Elastic  Limit 222 

177.  Examples  of  Existing  Structures 224 


APPENDIX  I. 

ON  THE  PRESERVATION  OF  WOOD. 

APPENDIX  II. 

TABLE  OF  THE  PROPERTIES  OF  MATERIALS. 


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A  TEEATISE 


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THE  RESISTANCE  OF  MATERIALS. 


INTEODUCTIOIS'. 


Library* 

Califc=rma 


1  .    IN    PROPORTIONING    ANY    MECHANICAL    STRUCTURE, 

there  are  at  least  two  general  problems  to  be  considered : — 

1st.  The  nature  and  magnitude  of  the  forces  which  are  to  be 
applied  to  the  structure,  such  as  moving  loads,  dead  weights, 
force  of  the  wind,  etc. ;  and, 

2d.  The  proper  distribution  and  magnitude  of  the  parts 
which  are  to  compose  the  structure,  so  as  to  successfully  resist 
the  applied  forces. 

These  problems  are  independent  of  each  other.  The  former 
may  be  solved  without  any  reference  to  the  latter,  as  the  struc- 
ture may  be  considered  as  composed  of  rigid  right  lines.  The 
latter  depends  principally  upon  the  mechanical  properties  of 
the  materials  which  compose  the  structure,  such  as  their  strength, 
stiffness,  and  elasticity,  under  various  circumstances. 

The  mechanical  properties  of  the  principal  materials — wood, 
stone,  and  iron  * — have  been  determined  with  great  care  and 
expense  by  different  experimenters,  both  in  this  and  foreign 
countries,  to  which  reference  will  hereafter  be  made. 

*  The  properties  of  mortars  have  been  thoroughly  discussed  by  Gen.  Q.  A. 
Gilmore  in  his  work  on  Limes,  Mortars,  and  Cements.    1862. 
1 


2  THE    RESISTANCE    OF   MATERIALS. 

2,    DEFINITIONS    OF    TERU1S. 

STRESSES  are  the  forces  which  are  applied  to  bodies  to  bring 
into  action  their  elastic  and  cohesive  properties.  These  forces 
cause  alterations  of  the  forms  of  the  bodies  upon  which  they 
act. 

STRAIN  is  a  name  given  to  the  kind  of  alterations  produced  by 
the  stresses.  The  distinction  between  stress  and  strain  is  not 
always  observed ;  one  being  used  for  the  other.  One  of  the 
definitions  given  by  lexicographers  for  stress,  is  strain  •  and  in- 
asmuch as  the  kind  of  distortion  at  once  calls  to  mind  the 
manner  in  which  the  force  acts,  it  is  not  essential  for  our  pur- 
pose that  the  distinction  should  always  be  made. 

A  TENSILE  STRESS,  or  Pull,  is  a  force  which  tends  to  elongate 
a  piece,  and  produces  a  strain  of  extension,  or  tensile  strain. 

A  COMPRESSIVE  STRESS,  or  Push,  tends  to  shorten  the  piece, 
and  produces  a  compressive  strain. 

TRANSVERSE  STRESS  acts  transversely  to  the  piece,  tending  to 
bend  it,  and  produces  a  bending  strain.  But  as  a  compressive 
stress  sometimes  causes  bending,  we  call  the  former  a  transverse 
strain,  for  it  thus  indicates  the  character  of  the  stress  wrhich 
produces  it.  Seams  are  generally  subjected  to  transverse 
strains. 

TORSIVE  STRESS  causes  a  twisting  of  the  body  by  acting  tan- 
gentially,  and  produces  a  torsive  strain. 

LONGITUDINAL  SHEARING  STRESS,  sometimes  called  a  detru- 
sive  strain,  acts  longitudinally  in  a  fibrous  body,  tending  to  draw 
one  part  of  .a  solid  substance  over  another  part  of  it ;  as,  for 
instance,  in  attempting  to  draw  the  piece  A  B, 
Fig.  1,  which  has  a  shoulder,  through  the  mortise 
C,  the  part  forming  the  shoulder  will  be  forced 
longitudinally  off  from  the  body  of  the  piece. 
so  that  the    remaining    part    may  be   drawn 
through. 

TRANSVERSE  SHEARING  STRES&  is  a  force  which  acts  trans- 
versely, tending  to  force  one  part  of  a  solid  body  over  the  adja- 


INTRODUCTION. 

cent  part.  It  acts  like  a  pair  of  shears.  It  is  the  stress  which 
would  break  a  tenon  from  the  body  of  a  beam,  by  acting  per- 
pendicular to  the  side  of  the  beam  and  close  to  the  tenon.  It 
is  the  stress  which  shears  large  bars  of  iron  transversely,  so 
often  seen  in  machine-shops.  The  applied  and  resisting  forces 
act  in  parallel  planes,  which  are  very  near  each  other. 

SPLITTING  STRESS,  as  when  the  forces  act  normally  like  a 
wedge,  tending  to  split  the  piece. 

3.  THE    EFFECT    OF    THESE     STRESSES     IS    TWOFOLD  :- 

1st.  Within  certain  limits  they  only  produce  change  of  form  ; 
and,  2d,  if  they  be  sufficiently  great  they  will  produce  rupture, 
or  separation  of  the  parts ;  and  these  two  conditions  give  rise 
to  two  general  problems  under  the  resistance  of  materials,  the 
former  of  which  we  shall  call  the  problem  of  ELASTIC  RESIST- 
ANCE ;  the  latter,  ULTIMATE  RESISTANCE,  or  RESISTANCE  TO 
RUPTURE. 

4.  GENERAL  PRINCIPLES  OF    ELASTIC   RESISTANCES.— 
(  To  determine  the  laws  of  elasticity  we  must  resort  to  experi- 
ment. )Bars  or  rods  of  different  materials  have  been  subjected 
to  different  strains,  and  their  effects  carefully  noted. 

From  such  experiments,  made  on  a  great  variety  of  materials, 
and  with  apparatus  which  enabled  the  experimenter  to  observe 
very  minute  changes,  it  has  been  found  that,  whatever  be  the 
physical  structure  of  the  materials,  whether  fibrous  or  granu- 
lar, they  possess  certain  general  properties,  among  which  are 
the  following : — 

1st.  That  all  bodies  are  elastic,  and  within  very  small  limits 
they  may  be  considered  perfectly  elastic  ;  i.  e.,  if  the  particles 
of  a  body  be  displaced  any  amount  within  these  limits  they  will, 
when  the  displacing  force  is  removed,  return  to  the  same  posi 
tion  in  the  mass  that  they  occupied  before  the  displacement 
This  limit  is  called  the  limit  of  perfect  elasticity* 

*  Mr.  Hodgkinson  made  some  experiments  to  prove  that  all  bodies  are  non- 
elastic.  (See  Civil  E 'ng.  and  Arch.  Jour.  vol.  vi. ,  p.  354.)  He  found  that  the 
limits  of  perfect  elasticity  were  exceedingly  small,  and  inferred  that  if  our 


THE   RESISTANCE   OF   MATERIALS. 


2d.  The  amount  of  displacement  within  the  elastic  limit  is 
directly  proportional  to  the  force  which  produces  it.  It  follows 
from  this,  that  in  any  prismatic  bar  the  force  which  produces 
compression  or  extension,  divided  by  the  amount  of  extension  or 
compression,  will  be  a  constant  quantity. 

3d.  If  the  displacement  be  carried  a  little  beyond  this  limit 
the  particles  will  not  return  to  their  former  position  when  the 
displacing  force  is  removed,  but  a  part  or  all  of  the  displace- 
ment will  be  permanent.  This  Mr.  Hodgkinson  called  a  set,  a 
term  which  is  now  used  by  all  writers  upon  this  subject. 

4th.  The  amount  of  displacement  is  not  exactly,  but  nearly, 
proportional  to  the  applied  force  considerably  beyond  the  elastic 
limit. 

5th.  Great  strains,  producing  great  sets,  impair  the  elasticity. 


5.    COEFFICIENT    (OR    MODULUS*)    OF    ELASTICITY. 

If  a  prismatic  bar,  whose  section  and  length  are 
unity,  be  compressed  or  elongated  any  amount  with- 
in the  elastic  limit,  the  quotient  obtained .  by 
dividing  the  force  which  produces  the  displacement 
by  the  amount  of  compression  or  extension  is  called 
the  COEFFICIENT  OF  ELASTICITY.  This  we  call  E. 
Let  K= section  of  a  prismatic  bar  (See  Fig.  2), 
Z=its  length, 

FIG.  2. 

powers  of  observation  were  perfect  in  kind  and  infinite  in  degree,  we  should 
find  that  no  body  was  perfectly  elastic  even  for  the  smallest  amount  of  dis- 
placement. And  although  more  recent  experiments  have  indicated  the  same 
result  in  cast-iron,  yet  the  most  delicate  experiments  have  failed  to  thoroughly 
establish  it.  I  have,  therefore,  accepted  the  principle  of  perfect  elasticity, 
which,  for  the  purposes  of  this  work,  is  practically,  if  not  theoretically,  correct. 
It  does  not  appear  from  Mr.  Hodgkinson's  report  how  soon  the  effect  was 
observed  after  the  strain  was  removed.  If  he  had  allowed  considerable  time 
the  set  might  have  disappeared,  as  it  is  evident  that  it  takes  time  for  the  dis- 
placed particles  to  return  to  their  original  position. 

*  The  terms  coefficient  and  modulus  are  used  indiscriminately  for  the  con- 
stants which  enter  equations  in  the  discussion  of  physical  problems,  and  are 
sometimes  called  physical  constants.  The  modulus  of  elasticity,  as  used  by 
most  writers  on  Analytical  Mechanics,  is  the  ratio  of  the  force  of  restitution  to 


INTRODUCTION.  5 

and  A=the  elongation  or  compression  caused  by  a  force,  P, 
which  is  applied  longitudinally.     Then 

T> 

K= force  on  a  unit  of  section,  and 

*=the  elongation  or  compression  for  a  unit  of  length. 

Hence,  from  the  definition  given  above,  we  have 
P       p         x         P/  m 

**  —  i  ~  T  —   ^x  '    V1) 

From  this  equation  E  may  be  easily  found.  It  will  here- 
after be  shown  that  the  coefficient  is  not  exactly,  but  is  nearly 
the  same  for  compression  as  for  tension. 

For  values  of  E,  see  Appendix,  Table  1. 

6.    PROOFS  OF    THE    I.AWS    GIVEN    IN    ARTICLE    FOUR.— 

Article  5  has  preceded  these  proof  s,  so  as  to  show  how  the  results 
of  experiments  may  be  reduced  by  equation  (1).  The  1st  and  2d 
laws  seem  first  to  have  been  proved  by  S.  Gravesend,  since  which 
they  have  been  confirmed  by  numerous  experimenters.  One  of 
the  most  extensive  and  reliable  series  of  experiments  upon  various 
substances  for  engineering  purposes  is  given  in  "  The  Report  of 
Her  Majesty's  Commissioners,  made  under  the  direction  of  Mr. 
Eaton  Hodgkinson."  The  results  of  his  experiments  are  pub- 
lished in  the  Reports  of  the  British  Association,  and  in  the  5th 
volume  of  the  Proceedings  of  the  Manchester  Literary  and 
Philosophical  Society,  from  which  extracts  have  been  made 
and  to  which  we  shall  have  occasion  to  refer.  The  experiments 
were  made  not  only  to  prove  these  laws  but  several  others, 
principally  relating  to  transverse  strength. 

Barlow  made  many  experiments,  the  results  of  which  are  given 
in  his  valuable  work  on  the  "  Strength  of  Materials."  The  series 
of  experiments  on  iron  which  had  been  commenced  and  so  ably 

that  of  compression.  It  relates  to  the  impact  of  bodies,  and,  as  thus  defined, 
depends  upon  the  set.  But  the  coefficient  of  elasticity  depends  neither  upon 
impact  nor  set.  Another  term  should  therefore  be  used,  or  else  a  distinction 
should  be  made  between  the  terms  coefficient  and  modulus,  so  that  the  former 
shall  apply  to  small  displacements,  and  the  latter  to  the  relative  force  of  resti- 
tution. For  this  reason  I  have  used  the  former  in  this  work,  and  avoided  the 
latter  when  applied  to  elasticity. 


6  THE    RESISTANCE    OF    MATERIALS. 

conducted  by  Mr.  Hodgkinson  were  continued  by  Mr.  Fairbairn. 
The  latter  confined  his  experiments  mostly  to  transverse  strength, 
the  results  of  which  are  given  in  his  valuable  work  on  "  Cast 
and  Wrought  Iron."  A  valuable  set  of  experiments  has  been 
made  in  France  at  "  le  Conservatoire  des  Arts  et  Metiers."  * 

In  this  country  several  very  valuable  sets  of  experiments  have 
been  made,  among  the  most  important  of  which  are  the  experi- 
ments of  the  Sub-Committee  of  the  Franklin  Institute,  the  re- 
sults of  which  are  published  in  the  19th  and  20th  volumes  of 
the  Journal  of  that  Society,  commencing  on  the  73d  page  of  the 
former  volume.  The  experiments  were  made  upon  boiler  iron, 
but  they  developed  many  properties  common  to  all  wrought 
iron.  They  were  conducted  with  great  care  and  scientific  skill. 
The  report  gives  a  description  of  the  testing  machine;  the 
manner  of  determining  its  friction  and  elasticity ;  the  modifica- 
tions for  use  in  high  temperature ;  the  manner  of  determining 
the  latent  and  specific  heats  of  iron  ;  and  the  strength  of  differ- 
ent metals  under  a  variety  of  circumstances. 

Another  very  valuable  set  of  experiments  was  made  by  Cap- 
tain T.  J.  Rodman  and  Major  W.  Wade,  upon  "Metals  for 
Cannon,  under  the  direction  of  the  United  States  Ordnance 
Department,"  and  published  by  order  of  the  Secretary  of  War. 

Numerous  other  experiments  of  a  limited  character  have 
been  made,  too  many  of  which  have  been  lost  to  science  be- 
cause they  were  not  reported  to  scientific  journals,  and  many 
others  were  of  too  rude  a  character  to  be  very  valuable. 

The  results  of  these  experiments  will  form  the  basis  of  our 
theories  and  analysis. 

*  See  "Morin's  Resistance  des  Materiaux,"  p.  126. 


EXPERIMENTS    ON   WROUGHT   IRON. 


CHAPTER    I. 

- 

TENSION. 

7.    TAKING    THE    PHENOMENA    IN  THEIR   NATURAL,    OR- 

DEH,  the  first  thing  which  claims  attention  is  the  elastic  re- 
sistance due  to  tension,  or,  as  it  is  sometimes  called,  a  pull,  or 
elongating  force. 

EXPERIMENTS    ON   WROUGHT   IRON. 

Experiments  for  determining  the  total  elongation  and  permanent  elongation  pro- 
duced by  different  ID  eights  acting  by  extension  on  a  tie  of  wrought  iron  of 
the  best  quality,  by  Eaton  Hodgkinson. 


Weight  in 

Elongation  per 

metre  of  length. 

Coefficient    of 

centimetre. 
P. 

Total.    A. 

Permanent. 

per  square  metre. 
E. 

Kil. 

187  429 

M. 

0  000082117 

Mill. 

Kil. 

22  824  500  000 

374  930 

0.000185261 

20  216  200  000 

502.406 
749.456 
937.430 
1124.813 
1312.283 
1499.720 
1687.219 
1874.645 
2063.580 
2249  627 

0.000283704 
0.000379467 
0.000475113 
0.000570792 
0.000665647 
0.000760311 
0.000873265 
0.001012911 
0.001283361 
0  002227205 

0.00254 
0.0033894 
0.0042398 
0.00508 
0.0067705 
0.0100879 
0.0330283 
0.0829955 
0.2616950 

19  824  100  000 
19  704  000  000 
19  729  909  000 
19  706  000  000 
19  714  600  000 
19  320  300  000 
19  320  700  000 
18398  100000 
16  079  200  000 

2403.653 
2824.564 

0.004287185 
0.009156490 
0.009950970 

3.0709900 
8.4690700 

8.5748700 

5  606  590  000 
2  866  380  000 

2812.033 
Repeated  after  1  hour 

0.010492805 
0.011750313 

9.1023600 

2  681  520  000 

'      2 

0.011858889 

'            '      3 

0  011933837 

'     4 

0011942168 

1      5 

0  011958835 

'      6 

0  011967149 

'      7 

0  012027114 

'      8 

0  012027014 

THE   RESISTANCE    OF   MATERIALS. 
EXPERIMENTS    ON   WROUGHT   IRON. Continued. 


Weight  in 

Elongation  per  i 

netre  of  length. 

Coefficient  of 
elasticity 

centimetre. 
P. 

Total.    A. 

Permanent. 

per  square  metre. 
E. 

Kil. 

Repeated  after  9  hours. 

0.012027114 

Mill. 

Kil.    , 

"          "  10      " 

0  012027114 

2999.500 
2999.500 

0.017888263 
0.019478898 

16.5145 

1  676  820  000 

0.01984831 

18.4212 

0.02022006 

18.8886 

3186.  97S 

0.02148590 
0  02169401 

19.7954 

1  483  290  000 

0.02170242 

0.02170242 

22  0119 

3374.440 

0.02477441 
0  02514184 

22.7087 

1  362  020  000 

0  02522512 

3561.900 

0.03493542 
0.03519357 

32.8201 

1  019  580  000 

0.03520190 

3745  361 

This  table  is  given  in  French  units  because  it  was  more  con- 
venient.* 

8.  THE  RESULTS  OF  THESE  EXPERIMENTS  may  be  re- 
presented graphically  by  taking,  as  has  been  done  in  Fig.  3.  the 
total  elongations  or  the  permanent  elongations  for  abscissas  and 
the  weights  for  ordinates. 

*  To  reduce  the  French  measures  to  English  we  have  the  following  relations : — 
LINEAR  MEASURE. 

3.2808992  feet  =  1  metre. 
0.0328089  feet  —  1  centimetre. 
0.0032808  feet  =  1  millimetre. 
0.0393696  in.     —  1  millimetre.. 

WEIGHT. 

2.20462  Ibs.  avoird.        =  1  kilogramme. 

1422.28  Ibs.  pr.  sq.  in.  =  1  kilog.  to  the  sq.  millimetre. 

0.0014S228  Ibs.  sq.  in.   —  1  kilog.  pr.  sq.  metre. 

Hence  to  reduce  the  above  quantities  to  English  units,  multiply  the  numbers 
in  the  first  column  by  14.2228  to  reduce  them  to  pounds  avoirdupois  per  square 
inch;  those  in  the  second  column  by  3.28089+  to  reduce  them  to  feet;  the 
third  by  0.03936+  to  reduce  them  to  inches  ;  and  the  fourth  by  0.00142228  to 
reduce  them  to  pounds  per  square  inch. 


TENSION. 


9 


When  the  construction  is  made  on  a  large  scale  it  makes  the 
results  of  the  experiments  very  evident. 


-TV.!!      UAaillllltt-     dOUU  j— 

tion   of    Fig.   3 

\ 

tint 

^ 

^1" 

shows  :  — 
1st    That  to  a 

J=i 

Jtt^, 

load  of  1499.72 

/? 

^^ 

kil.    pr.    square  "         // 
centimetre,    the  2000  [//-  

total  elongations           / 

are    practically 
proportional    to  J^QQ  |  

the  loads  ; 

2d.  That  with-    eo°  r 
in  the  same  lim-    Ann  1 

its    the    perma- 

nent elongations          0^  004l                 .Oi6                  .028          -OSG 
are   nearly  pro-                                     FIG.  3. 

portional  to  the  loads,  and  that  they  are  exceedingly  small ; 

3d.  That  beyond  the  load  of  14.997  kil.  to  22.00  kil.  per 
square  millimetre,  the  total  and  permanent  elongations  increase 
very  rapidly  and  more  than  proportional  to  the  loads ; 

4th.  That  near  and  beyond  22.49  kil.  per  square  millimetre, 
the  total  elongations  become  sensibly  proportional  to  the  loads, 
but  in  a  much  greater  ratio  than  that  which  corresponds  to  small 
loads.  For  the  loads  near  rupture  the  elongations  are  a  little 
inferior  to  that  indicated  by  the  new  proportion. 

5th.  Beyond  14.99  kil.  per  square  millimetre,  the  permanent 
elongations  increase  much  more  rapidly  than  the  total  elonga- 
tions. We  also  observe  that  the  permanent  elongations  increase 
with  the  duration  of  the  load,  although  very  slowly.  The  latter 

property  will  be  more  particularly  noticed  hereafter. 

p 
6th.  Finally,  the  values  —  of  the  loads  per  square  metre  to 

A 

the  elongation  per  metre,  and  which  is  called  the  coefficient  of 
elasticity,  is  sensibly  constant  when  the  elongations  are  nearly 
proportional  to  the  loads ;  and  that  the  mean  value  is 
E  —  19,816,440,000  kil.  per  square  metre ; 

=  28,283,000  Ibs.  per  square  inch. 
The  first  value  of  E,  in  the  table,  is  much  larger,  and  may 


10  THE    RESISTANCE    OF   MATERIALS. 

have  resulted  from  an  erroneous  measurement  of  the  exceed- 
ingly small  total  elongation.  From  the  experiments  made  on 
another  bar,  Hodgkinson  found 

E  =  19,359,458,500  kil.  per  sq.  metre ; 

=  27,VOO,000  pounds  pr.  square  inch  ; 
which  is  but  little  less  than  the  preceding. 

Mr.  Hodgkinson  infers  from  these  experiments  that  the  small- 
est strains  cause  a  permanent  elongation.  But  Morin  for- 
cibly remarks  *  that  none  of  these  experimenters  appear  to  have 
verified  whether  time,  after  the  strains  are  removed,  will  not 
cause  the  permanent  elongations  to  disappear.  Also  that  the 
deflections  of  the  machine  cannot  be  wholly  eliminated,  and 
hence  appear  to  increase  the  true  result.  In  practice  such  small 
permanent  elongations  may  be  omitted. 

The  preceding  example  has,  for  a  long  time,  been  given  to 
show  the  law  of  relation  between  the  applied  force  and  the 
total  and  permanent  elongations ;  but  we  should  not  expect  to 
find  exactly  the  same  results  for  all  kinds  of  iron.  Even  wrought 
iron  has  such  a  variety  of  qualities,  depending  upon  the  ore  of 
which  it  is  made,  and  the  process  of  manufacture,  that  it  cannot 
be  expected  that  the  above  results  will  always  be  applicable  to 
it.  •  Only  a  wide  range  of  experiments  will  determine  how  far 
they  may  generally  be  relied  upon. 

It  is  found,  however,  that  the  GENERAL  RESULTS  of  extension, 
of  set,  of  increased  elongation  with  the  duration  of  the  stress 
within  certain  limits,  and  of  the  increase  of  set  with  the  in- 
crease of  load,  are  true  of  all  kinds  of  iron. 

EXPERIMENTS    UPON    CAST   IRON. 
9.  THE  FOLLOWING  EXPERIMENTS  UPON  CAST  IRON  sllOW 

that  the  numerical  relation  between  the  applied  force  and  the 
extension  is  somewhat  different  from  the  preceding.  The  expe- 
riments were  made  under  the  supervision  of  Captain  T.  J.  Rod- 
man : —  f 

"  The  specimens  had  collars  left  on  them  at  a  distance  of  thirty-five  inches 


*  Morin's  Resistance  des  Materiaux,  p.  10. 

f  Experiments  on  Metals  for  Cannon,  by  Capt.  T.  J.  Rodman,  p.  157. 

For  a  full  description  of  the  testing  apparatus,  with  diagrams,  see  Major 
Wade's  Report  on  the  Strength  of  Materials  for  Cannon,  pp.  305-315.  The 
machine  consists  principally  of  a  very  substantial  frame  and  levers  resting  on 
knife  edges. 


TENSION. 


11 


apart,  the  space  between  the  collars  being1  accurately  turned  throughout  to  a 
uniform  diameter. 

"  The  space  between  the  collars  was  surrounded  by  a  cast-iron  sheath,  eight- 
tenths  of  an  inch  less  in  length  than  the  distance  between  the  collars ;  it  was 
put  on  in  halves  and  held  in  position  by  bands,  and  was  of  sufficient  interior 
diameter  to  move  freely  on  the  specimen. 

"  When  in  position,  the  lower  end  of  the  sheath  rested  on  the  lower  collar  of 
the  specimen,  the  space  between  its  upper  end  and  the  upper  collar  being  sup- 
plied with  and  accurately  measured  by  a  graduated  scale  tapered  0.01  of  an  inch 
to  one  inch. 

"  The  upper  end  of  the  sheath  was  mounted  with  a  vernier,  and  the  scale 
was  graduated  to  the  tenth  of  an  inch. 

"  This  afforded  means  of  measuring  the  changes  of  distance  between  the 
collars  to  the  ten-thousandth  part  of  an  inch,  and  these  readings  divided  by  the 
distance  between  the  collars  gave  the  extension  per  inch  in  length  as  recorded 
in  the  following  table  : — 

TABLE 

Sliowing  the  extension  and  permanent  set  per  inch  in  length  caused  by  the  under- 
mentioned weights,  per  square  inch  of  section,  acting  upon  a  solid  cylinder  35 
inches  long  and  1.366  inches  diametei'.  (Cast  at  the  West  Point  Foundry  in 
1857.) 


Weight  per 
square  inch  of 
section. 

Extension  per  inch  of 
length. 

Permanent  set  per  inch 
in  length. 

Coefficient  of 
elasticity. 

P. 

A. 

E. 

Ibs. 

in. 

in. 

1,000 

0.0000611 

0. 

16,366,612 

2,000 

0.0000794 

0. 

25,189,168 

3,000 

0.0001089 

0. 

27,548,209 

4,000 

0.0001771 

0. 

22,586,674 

5,000 

0.0002129 

0. 

23,489,901 

6,000 

0.0002700 

0.0000014 

22,222,222 

7,000 

0.0003328 

0.0000029 

21,033,653 

8,000 

0.0003986 

0.0000043 

20,070,245 

9,000 

0.0004557 

0.0000071 

19,749,835 

10,000 

0.0001100 

0.0000109 

19,607,843 

11,000 

0.0005500 

0.0000157 

20.000,000 

12,000 

0.0006414 

0.0000257 

18,693,486 

13,000 

0.0007100 

0.0000300 

18,309,859 

14,000 

0.0007700 

0.0000357 

18,181,181 

15,000 

0.0008557 

0.0000477 

17,529,507 

16,000 

0.0009243 

0.0000529 

17,310,397 

17,000 

0.0010014 

0.0000643 

16,977,231 

18,000 

0.0010900 

0.0001014 

16,537,614 

19,000 

0.0012271 

0.0001471 

15,483,660 

20,000 

0.0013586 

0.0002014 

14,721,109 

21,000 

0.0015386 

0.0002900 

13,648,771 

22,000 

0.0017043 

0.0003986 

12,908,523 

23,000 

0.0019529 

0.0005529 

11,265,246 

24,000 

0.0022786 

0.0007529 

10,532,344 

25,000 

0.0026037 

0.0010843 

9,601,720 

26,000 

0.0032186 

8,078,046 

12 


TfTE    RESISTANCE   OF   MATERIALS. 


1O.    FIGURE    4    IS    A    GRAPHICAL,    REPRESENTATION    OF 

THE  AHOVE  TABLE,  constructed  in  the  same  way  as  Figure  3. 
Experiments  were  made  upon  many  other  pieces,  from  which 
I  have  selected  four,  and  called  them  A,  B,  C,  and  D,  a  gra- 
phical representation  of  which  is  shown  in  Figure  5.  The  right 
hand  lines  represent  extensions,  the  left  hand  sets. 


FIG.  4. 


FIG.  5. 


A  is  from  an  inner  specimen  of  a  Fort  Pitt  gun,  No.  335, 
and  the  others  from  different  cylinders  which  were  cast  for  the 
purpose  of  testing  the  iron. 

From  these  we  observe : — 

1st.  That  for  small  elongations  the  ratio  of  the  stresses  to  the 
elongations  is  nearly  constant. 

2d.  There  does  not  appear  to  be  a  sudden  change  of  the  rate 
of  increase,  as  in  Mr.  Hodgkinson's  example,  but  the  ratio  gra- 
dually increases  as  the  strains  increase. 

3d.  The  sets  at  first  are  invisible,  but  they  increase  rapidly 
as  the  strains  approach  the  breaking  limit. 

It  appears  paradoxical  that  the  first  and  second  experiments 
in  the  preceding  table  should  give  a  less  coefficient  than  the 
third,  but  the  same  result  was  observed  in  several  cases. 

11.  THE  FOLLOWING  TABLES  ARE  THE  RESULTS  OF 
SOME  EXPERIMENTS  MADE.BY  MR.  HODGKINSON  :— 


TENSION.  13 

'  Direct  longitudinal  extension  of  round  rods  of  cast  iron,  fifty  feet  long. 


a 

2 

•£ 

Weights  per  square  inch  laid  on, 

£«« 

1  . 

•M    g 

'«  fl' 

with  their  corresponding  ex- 

"a ^""S 

.§  § 

NAME  OF   IRON. 

1 

§0 

tensions  and  sets. 

£S.| 

gi 

ft 

1 

a 

Weights. 

Extension. 

Sets. 

ii-g 
afi 

Ibs. 

in. 

in. 

Ibs. 

in. 

2 

1.058 

2,117 

0.0950 

0.00345 

16,408 

1.085 

6,352 

0.3115 

0.0250 

10,586 

0.5740 

0.4425 

14,821 

0.9147 

0.12775 

Blaneavon  Iron,  No.  2.  . 

2 

1.0685 

2,096 

0.0942 

0.00268 

14,675 

0.9325 

6,289 

0.3065 

0.01675 

10,482 

0.5770 

0.0575 

13,627 

0.8370 

0.11475 

Gartsherrie  Iron,  No.  2. 

2 

1.062 

2,109 

0.0922 

0.001  + 

16,951 

1.167 

6,328 

0.3117 

0.01450 

10,547 

0.5862 

0.0475 

14,766 

0.9452 

0.11352 

Jn  these  experiments  the  ratio  of  the  extensions  is  somewhat 
greater  than  that  of  the  weights.  The  value  of  E,  as  computed 
for  the  first  weights  which  are  given,  and  the  corresponding 
extensions,  is  a  little  more  than  13,000,000  pounds  per  square 
inch. 


Extension  of  cast-iron  rods,  ten  feet  long  and  one  inch  square. 

3S|^~< 
a  t»  8  g  ;- 

Weicrhts 
P. 

Extensions. 

Ae. 

Sets. 

p 

I*lilji 

Ibs. 

in. 

in. 

1053.77 

.0090 

.00022 

117086 

—     4^ 

1580.65 

.0137 

.000545 

115131 

—    aV 

2167.54 

.0186 

.00107 

113309 

-T}S 

3161.31 

.0287 

.00175 

110150 

4215.08 

.0391 

.00265 

107803 

+  7sT 

5262.85 

.0500 

.00372 

105377 

+  T63 

6322.62 

.0613 

.00517 

103142 

Htfr 

7376.39 

.0734 

.00664 

100496 

+    T25 

8430.16 

.0859 

.00844 

98139 

+  75<T 

9483.94 

.0995 

.01062 

95316 

+  7^T 

10537.71 

.1136 

.01306 

92762 

+  7^7 

11591.48 

.1283 

.01609 

90347 

-rhr 

12645.25 

.1448 

.02097 

87329 

-ik 

13699.83 

.1668 

82133 

+  *ta 

14793.10 

.1859 

!  02410 

79576 

-A 

14  THE   RESISTANCE    OF    MATERIALS. 

Let  P  =  the  elongating  force  and 

\  =  the  total  elongation  in  inches  due  to  P. 
Then  Hodgkinson  found,  from  an  examination  of  the  table, 
that  the  empirical  formula 

P  =  116im  -  201905> 


represented  the  results  more  nearly  than  equation  (1).  This  for- 
mula reduced  to  an  equivalent  ayg  for  I  in  inches  (observing 
that  the  bar  was  10  feet  long),  becomes 

i. 
P  =  13,934,000   -*  —  2,907,432,000.^ 

Although  this  equation  gives  the  elongations  for  a  greater  range 
of  strains  than  equation  (1)  for  this  particular  case,  yet  the 
law  represented  by  it  is  more  complicated,  and  hence  would 
make  the  discussions  under  it  more  difficult,  without  yielding 
any  corresponding  advantage.  It  is  the  equation  of  a  parabola 
in  which  P  is  the  abscissa  and  Ae  the  ordinate. 

We  also  see  that  when  the  elongations  are  very  small,  the 

A2 
quantity  -=~  will  be  very  small,  and  the  second  term  may  be 

i 

omitted  in  comparison  with  the  first,  in  which  case  it  will  be  re- 
duced to  equation  (1).  The  coefficient  in  the  first  term  is  the 
coefficient  of  elasticity,  hence  it  is  nearly  14,000,000  Ibs. 
for  extension. 


MALLEABLE   IRON. 

1£.  ACCORDING  TO  BARLOWS  EXPERIMENTS  malleable 
iron  may  be  elongated  T¥Vtr  of  its  length  without  endangering 
its  elasticity.*  To  ascertain  this,  the  strains  wTere  removed 
from  time  to  time,  and  it  was  found  that  the  index  returned  to 
zero  for  all  strains  less  than  9  or  10  tons.  The  mean  extension 
per  ton  (of  2,240  Ibs.)  per  square  inch,  for  four  experiments,  was 
0.00009565  of  its  original  length.  Hence  the  mean  value  of 
the  coefficient  of  elasticity  is 


E  =  —  =  _  --  _  =  23,418.000  Ibs. 
A        0.00009565 

*  Journal  Frank.  Inst.,  voL  xvi.,  2d  Series,  p.  126. 


TENSION.  15 

ELASTICITY    OF   WOOD. 
13.    EXPERIMENTS  BY  MESSRS.  CHEVANDIER  ANI*  WER- 

THEiM.-The  following  are  some  of  the  results  of  the  recent 
experiments  of  Messrs.  Chevandier  and  Wertheim  on  the  resis- 
tance of  wood.  These  experimenters  have  drawn  the  follow- 
ing principal  conclusions : — 

1.  The  density  of  wood  appears  to  vary  very  little  with  age. 

2.  The  coefficient  of  elastioky  diminishes,  on  the  contrary,  be- 

yond a  certain  age ;  it  depends,  likewise,  upon  the  dry- 
ness  and  the  exposure  of  the  soil  to  the  sun  in  which  the 
trees  have  grown  ;  thus  the  trees  grown  in  the  northern 
exposures,  north-eastern,  north-western,  and  in  dry  soils, 
have  always  so  much  the  higher  coefficient  as  these  two 
conditions  are  united,  whereas  the  trees  grown  in  muddy 
soils  present  lower  coefficients. 

3.  Age  and  exposure  influence  cohesion. 

4.  The  coefficient  of  elasticity  is  affected  by  the  soil  in  which 

the  tree  grows. 

5.  Trees  cut  in  full  sap,  and  those  cut  before  the  sap,  have  not 

presented  any  sensible  differences  in  relation  to  elasticity. 

6.  The  thickness  of  the  woody  layers  of  the  wood  appeared  to 

have  some  influence  on  the  value  of  the  coefficient  of 
elasticity  only  for  fir,  which  was  greater  as  the  layers  were 
thinner. 

7.  In  wood  there  is  not,  properly  speaking,  any  limit  of  elasti- 

ticity,  as  every  elongation  produces  a  set. 
It  follows  from  this  circumstance  that  there  is  no  limit  of 
elasticity  for  the  woods  experimented  upon  by  Messrs.  Chevan- 
dier and  Wertheim,  but  in  order  to  make  the  results  of  their  ex- 
periments agree  with  those  of  their  predecessors,  the  authors 
have  given  for  the  value  of  the  limit  of  elasticity  the  load  under 
which  it  produces  only  a  very  small  permanent  elongation  ;  the 
limit  which  they  indicate  in  the  following  table  for  loads  under 
which  the  elasticity  of  wood  commences  to  change,  corresponds 
to  a  permanent  elongation  of  0.00005,  its  original  length. 


16 


THE   RESISTANCE   OF   MATEEIALS. 


TABLE  CONTAINING  THE  MEAN  RESULTS  OF  THE  EXPERIMENTS  OF 
MESSRS.  ClIEVANDIER  AND  WERTHEIM. 


Species. 

I 

1  Coefficient  of  elas- 
ticity E  referred  to 
Ithe  square  milli- 
metre. 

Limit  of  elasticity, 
or  load  per  square 
millimetre,  corres- 
ponding to  that  li- 
mit. 

Cohesion,  or  load 
per  square  milli- 
metre capable  of 
producing  rupture. 

Val.  of  E  in  pounds 
per  square  inch. 

0  717 

Kilogr. 

1261.9 

Kilopr. 
3.188 

Kilogr. 

7  93 

£J> 

Fir                       

0.493 

1113.2 

2.153 

4.18 

Yoke  Elm  

0.756 

1085.3 

1.282 

2.99 

Birch  

0.812 

997.2 

1.617 

4.30 

0.823 

980.4 

2.317 

3.57 

.'§-§  g 

Oak  from  pedunculate  acorn 
"      "     sessile  acorn.  .  .  . 
White  Pine   

0.808 
0.872 
0.559 

977.8 
921.8 
564.1 

u 

2.349 
1.633 

6.49 
5.66 

2.48 

ill 
*H     % 

Elm  

0.723 

1165.3 

1.842 

6.99 

O  O   3> 
-toO    P< 

Sycamore      .  .  

0  692 

1163  8 

1.139 

6  16 

rt  O 

S3  _r  M 

Ash               

0697 

1121.4 

1.246 

6  78 

•s§£ 

Alder         

0601 

1108.1 

1.121 

454 

SB  °o 
ffii-Ts 

0.602 

1075.9 

1.035 

7.20 

o      o. 

0.674 

1021.4 

1.068 

3.58 

o  So 

Poplar.  . 

0.477 

517.2 

1.007 

1.97 

#£~- 

14.      ELASTICITY  OF  WOOD,   TANGENTIAL!,  Y  AND  RATI- 

The  same  observers  have  also  determined  the  coefficient 
of  elasticity  and  the  cohesion  of  wood  in  the  direction  of  the 
radius  and  in  the  direction  of  the  tangent  to  the  woody  layers. 
An  examination  of  the  following  Table  shows  that  the  resis- 
tance in  the  direction  of  the  radius  is  always  greater  than  the 
resistance  in  the  direction  of  the  tangent  to  the  woody  layers ; 
the  relation  between  the  coefficients  of  elasticity  in  the  two 
cases  varying  nearly  from  3  to  1.15. 


TENSION. 


MEAN  RESULTS  OF  THE  EXPERIMENTS  OF  MESSRS.  CHEVANDIER 
AND  WERTHEIM. 


[SPECIES. 

IN  THE  DIRECTION  OP  BADIUS. 

IN  THE  DIRECTION  OP  THE  TAN- 
GENT TO  THE  LA.YERS. 

Coefficient    of 
Elasticity,  E,  per 
square    millime- 
tre. 

Cohesion,      or 
load,  per  square 
millimetre,  capa- 
ble of  producing 
rupture. 

Coefficient    of 
Elasticity,  E,  per 
square    millime- 
tre. 

Cohesion,      or 
load,  per  square 
millimetre,  capa- 
ble of  producing1 
rupture. 

Yoke  Elm     

Kilogr. 

208.4 
134.9 
157.1 
188.3 
81.1 
269.7 
111.8 
321.6 
94.5 
97.7 
170.3 

Kilogr. 

1.007 
0.522 
0.716 
0.582 
0.823 
0.885 
0.218 
0.345 
0.220 

0.256 
K 

Kilogr. 

103.4 
80.5 
72.7 
129.8 
155.2 
159.3 
102.0 
63.4 
34.1 
28.6 
152.2 

Kilogr. 
,   0.608 

0.610 
0.371 
0.406 
1.063 
0.752 
0.408 
0.366 
0.297 
0.196 
1.231 

Sycamore  

Maple  .  . 

Oak 

Birch  

Beech  

Ash  

Elm 

Fir    

Pine  

Locust  .  . 

The  highest  coefficient  of  elasticity  in  this  table  is  for  beech,  and  this  is  less, 
than  400,000  pounds  per  square  inch. 


15.  REMARK. — The  value  ofE,  which  is  used  in  practice,  i& 
not  the  coefficient  of  perfect  elasticity,  but  it  is  that  value  which 
is  nearly  constant  for  small  strains.    In  determining  it,  no  ac- 
count is  made  of  the  set.     If  the  total  elongations  were  propor- 
tional to  the  stresses  which  produce  them,  we  would  use  the 
value  of  E  found  by  them,  even  if  the  permanent  equalled  the 
total  elongations.      But  in  practice  the  permanent  elongations 
will  be  small  compared  with  the  total  for  small  stresses. 

APPLICATIONS. 

16.  TO    FIND    THE    ELONGATION    OF     A  PRISMATIC   BAR 
SUBJECTED  TO  A   LONGITUDINAL   STRAIN  WHICH   IS  WITH- 
IN THE  ELASTIC  LIMITS. 

From  (1)  we  have 


_ 
EK 


(2) 


which  is  the  required  formula. 

2 


18 


THE    RESISTANCE    OF   MATERIALS. 


Also  from  (1)  we  have 


P  =  7EK 


(3) 


FIG.  6. 


Equations  (1),  (2),  and  (3)  are  equally  applicable 
to  compressive  strains,  as  will  hereafter  be  shown. 
If  in  (3)  we  make  K=l  and  A— I  we  shall  have 
P  =  E  ;  hence,  the  coefficient  of  elasticity  may  be  de- 
fined to  be  a  force  which  will  elongate  a  bar  whose 
section  is  unity,  to  double  its  original  length,  pro- 
vided the  elasticity  of  the  material  does  not  change. 
But  there  is  no  material,  not  even  a  perfectly  elastic 
body, — as  air  and  other  gases, — whose  coefficient  of 
elasticity  will  not  change  for  a  perceptible  change  of  volume. 
The  material  may  not  lose  its  elasticity,  but  equation  (1)  only 
measures  it  for  small  displacements.  To  illustrate  further,  let 
it  be  observed  that,  according  to  Mariotte's  law,  the  volumes  of 
a  gas  are  inversely  proportional  to  the  compressive  (or  exten- 
sive) forces ;  double  the  force  producing  a  compression  of  half 
the  volume  ;  four  times  the  force,  one-fourth  the  volume,  and 
so  on, — the  compressions  being  a  fractional  part  of  the  original 
volume  ;  but  in  equation  (2),  A  is  a  linear  quantity,  so  that  if 
one  pound  produces  an  extension  (or  compression)  of  one  inch, 
two  pounds  would  produce  an  extension  of  two  inches,  and 
so  on. 


. 


Examples.— -1.  If  the  coefficient  of  elasticity  of  iron  be  25,000,000  Ibs., 
what  must  be  the  section  of  an  iron  bar  60  feet  long,  so  that  a  weight  of  5,000 
Ibs.  shaU  elongate  it  i  inch  ? 

PI 
From  (1)  we  obtain  K  =  —  which  by  substitution  becomes 


K  = 


5,000.12  x  60 
25^00^000^ 


/ 
—  0.288  square  inches. 


2.  How  great  a  weight  will  a  brass  wire  sustain,  whose  diameter  is  1  inch  ; 
•coefficient  of  elasticity  is  14,000,000  Ibs.,  without  elongating  it  more  than  -g-tn; 
of  its  length?  Ans.  13,744.5  Ibs. 


TENSION. 


19 


*  «  •    REQUIRED  THE  ELONGATION  (OR  COMPRESSION)  OF 
A  PRISMATIC  BAR    WHEN  ITS  WEIGHT  IS  CONSIDERED. 

Let  I  =  the  whole  length  of  the  bar  before  elon- 
gation or  compression, 

x  =  variable  distance  =  AJ, 

dx  =  1)0  —  an  element  of  length, 

w  =  weight  of  a  unit  of  length  of  the  bar, 

W  =  weight  of  the  bar,  and 

Pj  =  the  weight  sustained  by  the  bar. 
Then  (I  —  x)  w  -f-  P*  =  P  =  the  strain  on  any 
section,  as  be. 

Hence,  from  equation  (2),  we  have  PlG  7 


x  = 


EK 


_ 


_IV 


EK 


-     (4) 


the  total  length  will  become, 


If 


I 


(5) 


or 


one- 


half  of  what  it  would  be  if  a  weight  equal  to  the  whole  weight 
of  the  bar  were  concentrated  at  the  lower  end. 

REQUIRED  THE  ELONGATION  (OK  COMPRESSION)  OF  A  CONE  IN  A 
VERTICAL  POSITION,  CAUSED  BY  ITS  OWN  WEIGHT  WHEN  IT  is  SUS- 
PENDED AT  ITS  BASE  (OR  RESTS  ON  ITS  BASE). 

Take  the  origin  at  the  apex  before     '/////////////////. 
extension,    Fig.     8,    and 
let  K  =  any  section, 

K0  =  the  upper  secti®n, 

I  =  the  length  or  altitude  of 

the  cone, 

x  —-  the  length  or  altitude  of 
any  portion  of  the  cone, 
and 

8  =  the  weight  of  a  unit  of  volume. 
Then,  because  the  bases  of  similar  cones  are  as  the  squares 


FIG.  8. 


of  their  altitudes,  K  =  K0  - 


20  THE    RESISTANCE    OF   MATERIALS. 

The  volume  of  the  cone  whose  altitude  is  x 


/ 


and  the  weight  of  the  same  part 


.-.  (from  equation  (2)  )  \  = 


from  which  it  appears  that  the  total  elongation  is  independent 
of  the  transverse  section,  and  varies  as  the  square  of  the  length. 

1  8.  THE  WORK  OF  ELONGATION.  —  If  P  be  the  force  which 
does  the  work,  and  x  the  space  over  which  it  works,  then  the 
general  expression  for  the  work  is 


(6) 


To  apply  this  to  the  prism,  substitute  P  from  Eq.  (3)  in  (6), 
and  make  dx  =  d\  and  we  have 


which  is  the  same  result  that  we  would  have  found  by  suppos- 
ing that  P  was  put  up  by  increments,  increasing  the  load  gradu- 
ally from  zero  to  P. 

Example. — If  the  coefficient  of  elasticity  of  wrought  iron  be  28,000,000  Ibs., 
and  is  expanded  0. 00000698  of  its  length  for  one  degree  F. ,  how  much  work  is 
done  upon  a  prismatic  bar  whose  section  is  one  inch,  and  length  30  feet,  by  a 
change  of  20  degrees  of  temperature  ? 

Walls  of  buildings  which  were  sprung  outward  have  been  drawn  into  an  erect 
position  by  heating  and  cooling  bars  of  iron.  Several  rods  were  passed  through 
the  buildings,  and  extending  from  wall  to  wall,  were  drawn  tight  by  means  of 
the  nuts.  Then  a  part  of  them  were  heated,  thus  elongating  them,  and  the 
nuts  tightened;  after  which  they  were  allowed  to  cool,  and  the  contraction 
which  resulted  drew  the  walls  together.  Then  the  other  rods  were  treated  in 
a  similar  manner,  and  so  on  alternately. 


TENSION. 


19.  VERTICAL  OSCILLATIONS — If  a  bar  Aa,  Fig.  9, 
with  a  weight,  P,  suspended  from  its  lower  end,  be  pressed  down 
by  the  hand,  or  by  an  additional  weight  from  a  to  #,  and  the  addi- 
tional force  be  suddenly  removed,  the  end  of  the  bar  on  returning 
will  not  stop  at  «,  but  will  move  to  some  point  above,  as  c,  a  dis- 
tance ac  =  ad.  From  a  principle  in  Mechanics,  viz. ,  that  the  living 
force  equals  twice  the  work,  we  are  enabled  to  determine  all  the 
cireumstances  of  the  eseillation  when  the  weight  of  the  bar  is 
neglected.  The  weight  P  elongates  the  bar  so  that  its  lower  ex- 
tremity is  at  «,  at  which  point  we  will  take  the  origin  of  co-or- 
dinates. 


Fig.  9. 
Let  A  =  ab  =  the  elongation  caused  by  the  additional  force, 

x  =  ad  —  any  variable  distance  from  the  origin, 

v  =  the  velocity  at  any  point,  as  d,  and 

M  —  the  mass  of  the  weight  P. 
If  the  weight  of  the  rod  be  very  small  compared  with  P,  the  vis  viva  is 

P    ,  W^txvi 

M»2  =  —  «2  very  nearly. 

y 

EK 

The  work  for  an  elongation  equal  to  A,  is  by  Eq.  (7),  -^r 


P  EK  da?  i-"   EK 

Mi   *2  *  "     ^-^  or  - 


/~K~      r       d 
~V  *EK    /     ^== 

J 


dx  P          _>Z-IA      , 

====\/^  s        T_|O  =2 


J 


^ 


for  half  an  oscillation  ;  and  the  time  for  a  whl^olkjjtfation  is 


fcJ- 


_ 

hence  the  oscillations  will  be  isochronous,  v^  &  OIA*A..**>  I  ^ 

It  is  evident  that  by  applying  and  removing  the  force  at  regular  intervals,  the 

amplitude  of  the  oscillations  may  be  increased  and  possibly  produce  rupture. 

In  this  way  the  Broughton  suspension  bridge  was  broken.* 

As  a  second  example  take  the  case  in  which  P  is  applied  suddenly  to  the  end 

of  the  rod.     It  is  evident  that  the  total  elongation  will  be  greater  than  X,  —  the 

permanent  elongation.     For  the  fundamental  equation  we  may  use  another 


*  Mr.  E.  Hodgkinson,  in  the  4th  volume  of  the  Manchester  Philosophic  il 
Transactions,  gives  the  circumstances  of  the  failure,  from  this  cause,  of  the 
suspension  bridge  at  Broughton,  near  Manchester,  England.  And  M.  Navier,  in 
his  theory  of  suspension  bridges  (Fonts  Suspendus,  Paris,  1823),  states  that 
the  duration  of  the  oscillation  of  chain  bridges  may  be  nearly  six  seconds. 


«  ^L 


' 


/Cc^&c 


4  . 


22  THE   RESISTANCE    OF   MATEEIALS. 

' 

principle  in  Mechanics,  which  might  have  been  used  in  the  preceding  problem, 
viz.,  that  the  mass  multiplied  by  the  acceleration  equals  the  moving  force.  The 

EKz 

resisting  force  for  an  elongation  x  is  —7—  (See  Eq.  (3)  ),  and  the  moving  force  is 

P 

P,  whose  mass  =  —  ;  hence 


Mversin     '  5g* 


Ifx^  A,     V=   V^A, 

a;  =  2A,  ®  =       0, 
*  =  frt    0  =       0. 
Hence,  the  amplitude  is  twice  the  permanent  elongation.     If  x  —  2A  we  have 

t  =  IT  */  — j££  =  ir    /  -—.     Investigations  of  this  kind  give  rise  to  a  divi- 


sion of  the  subject  called  Resilience  of  Prisms. 

The  investigations  are  interesting,  but  the  results  are  of  little  use  beyond 
those  which  have  already  been  indicated.  From  the  last  problem  we  see  that  a 
weight  suddenly  applied  produces  twice  the  strain  that  it  would  if  applied 
gradually. 

As  additional  exercises  for  the  student,  I  suggest  the  following  :  Suppose  the 
weight  be  applied  with  an  initial  velocity.  Suppose  a  weight  P  is  attached  to 
one  end,  and  the  weight  P'  is  placed  suddenly  upon  it ;  or  it  falls  upon  it. 
To  find  the  velocity  at  any  point  in  terms  of  £,  —  also  A  in  terms  of  t. 

If  a  weight  W  is  suspended  at  the  end,  and  another  weight  Wi  falls  from  a 
height  7i,  giving  rise  to  a  velocity  #,  we  have  for  the  common  velocity  of  the 

bodies  after  impact,  if  both  are  non-elastic,  V  = 1^_ ,  and  the   ?  is  viva  of 

both  will  be 

"W  2 tfi  "^FTC" 

MV2  =     - which  equals  _ —  A2,  or  twice  the  work. 

_Wi_       I~U_ 
This  is  only  an  approximate  value,  for  the  inertia  of  the  wire  is  neglected. 

2O.  VISCOSITY  OF  SOLIDS. — Experiments  show  that  the  prin- 
ciple of  equal  amplitudes,  referred  to  in  the  preceding  article,  is 
not  realized  in  practice.  This  is  more  easily  observed  in  trans- 
verse vibrations.  The  amplitudes  grow  rapidly  less  from  the 
first  vibration,  and  the  diminution  cannot  be  fully  accounted  for 
by  the  external  resistance  of  air.  Professor  Thompson  of  Eng- 


TENSION.  23 

land  has  shown  that  there  is  an  internal  resistance  which  opposes 
motion  among  the  particles  of  a  body,  and  is  similar  to  that 
resistance  in  fluids  which  opposes  the  movement  of  particles 
among  themselves.  He  therefore  called  it  viscosity.*  He  proved : 

1st.  That  there  was  a  certain  internal  resistance  which  he 
called  Viscosity,  and  which  is  independent  of  the  elastic  pro- 
perties of  nintfilfr»  ?>v4/t 

2d.  That  this  force  does  not  affect  the  co-efficient  of  elasticity. 

The  law  between  molecular  friction  and  viscosity  was  not 
discovered. 

The  viscosity  was  always  much  increased  at  first  by  the  in- 
crease of  weight,  but  it  gradually  decreased,  and  after  a  few 
days  became  as  small  as  if  a  lighter  weight  had  been  applied. 
Only  one  experiment  was  made  to  determine  the  effect  of  con- 
tinual vibration ;  and  in  that  the  viscosity  was  very  much  in- 
creased by  daily  vibrations  for  a  month. 

This  latter  fact,  if  firmly  established,  will  prove  to  be  highly 
important ;  for  it  shows  that  materials  which  are  subjected  to 
constant  vibrations,  such  as  the  materials  of  suspension  bridges, 
have  within  themselves  the  property  of  resisting  more  and  more 
strongly  the  tendency  to  elongate  from  vibration.  Experi- 
ments will  be  given  hereafter  which  tend  to  confirm  this  fact, 
when  the  vibrations  are  not  too  frequent  or  too  severe. 

But  the  true  viscosity  of  solids  has  been  fully  proved  by  Mr. 
Tresca,  a  French  physicist,  who  showed  that  when  solids  are 
subjected  to  a  very  great  force,  the  amount  of  the  force 
depending  upon  the  nature  of  the  material,  that  the  particles  in 
the  immediate  vicinity  of  pressure  will  jf<w0  over  each  other,  so 
as  to  resemble  the  flowing  of  molasses,  or  tar,  or  other  viscous 
fluids.  Thus,  the  true  viscosity  differs  entirely  in  its  character 
from  the  property  recognized  by  Professor  Thompson. 


RESISTANCE  TO  RUPTURE  BY  TENSION. 
21.  MODUL.US  OF  STRENGTH. — Many  more  experiments 
have  been  made  to  determine  the  ultimate  resistance  to  rupture 
by  tension,  than  there  have  to  determine  the  elastic  resistance. 
In  the  earlier  experiments  the  former  was  chiefly  sought,  and 
more  recently  all  who  experimented  upon  the  latter  also  deter- 
mined the  former. 


*  Civ.  Eng.  Jour. ,  vol. 


THE   RESISTANCE    OF   MATERIALS. 

The  force  which  is  necessary  to  pull  asunder  a  prismatic  bar 
whose  section  is  one  square  inch,  when  acting  in  the  direction 
of  the  axis  of  the  bar,  is  called  the  mod/ulus  of  strength.  This 
we  call  T.  It  expresses  the  tenacity  of  the  material,  and  is 
sometimes  called  the  absolute  strength  and  sometimes  modulus 
of  tenacity. 

&2.    FORMULA  FOR  THE  MODULUS  OF  STRENGTH;  OT  the 

force  necessary  to  break  a  prismatic  bar,  when  acted  upon  by 
a  tensile  strain. 

Let  K— the  section  of  the  bar  in  inches, 
T— the  modulus  of  tenacity,  and 
P=the  required  force. 

It  is  proved  by  experiment  that  the  resistance  is  proportional 
to  the  section  ;  hence 

P=TK (9) 

.'.  T=|  .      (10) 

""From  (10)  T  may  be  found.     In  (10)  if  P  is  not  the  ultimate 
resistance  of  the  bar,  then  will  T  be  the  strain  on  a  unit  of  section. 
From  (9)  we  have 

K=J  (11) 

which  will  give  the  section. 

The  following  are  some  of  the  values  of  T  which  have  been 
found  from  experiment  by  the  aid  of  equation  (10). 

Cohesive  force  or  Tenacity 
in  pounds  per  square  inch. 

Ash  (English)        .         .         .         .         .  17,000 

Oak  (English) 9,000  to    15,000 

Pine  (pitch) 10,500 

Cast  Iron* 14,800  to    16,900 

Cast  Iron  ( Weisbach  <&  Overman)          .  20,000 

Wrought  Iron 50,000  to    65,000 

Steel  wire 100,000  to  120,000 

Bessemer  steel  f 120,000  to  129,000 

"          "$.....  72,000  to  101,000 

Bars  of  Crucible  Steel  §     ....  70,000  to  134,000 

The  most  remarkable  specimen  of  cast  steel  for  tenacity  which 

*  Hodgkinson,  Bridges.     Weale,  sup.,  p.  25. 
f  Jour.  Frank.  Inst.     Vol.  84,  p.  366. 

\  Also  experiments  by  Wm.  Fairbairn,  Van  Nostrand's  EC.  En.  Mag.,  Vol. 
I.,  p.  273.  §  Do.  p.  1009. 


TENSION.  25 

is  on  record,  was  manufactured  in  Pittsburgh,  Pa.  It  was 
tested  at  the  Navy  Yard  at  Washington,  D.  C.,  and  was  found 
to  sustain  242,000  pounds  to  the  square  inch  !  * 

For  other  values  see  the  Appendix. 

23.  A  vertical  prismatic  bar  is  fixed  at  its  upper  end,  and 
a  weight  Pl  is  suspended  at  the  other  /  what  must  be  the  upper 
section,  at  A,  Fig.  7  ,  so  as  to  resist  n  times  all  the  weight  below 
it,  the  weight  of  the  bar  being  considered? 

Let  <?  =  the  weight  of  a  unit  of  volume  of  the  bar,  and  the 
other  notation  as  before. 


/.     K  =  n*i-     ...........     (12) 

T-ntl  '      •  } 

p 

If  n  =  1,  K  =  j^-^-  ;  and  if  $  I  =  T,  K=  oo,  or  no  ^section  is  pos- 


T 

sible,  and  I  =  _  is  the  corresponding  length  of  the  bar. 


BAR  OF  UNIFORM  STRENGTH.      Suppose  a  bar  is 

fixed  at  its  upper  extremity,  Fig.  1O?  and  a  weight  P^  is  sus- 
pended at  its  lower  extremity;  it  is  required  to  find  the  form 
of  the  bar  so  that  the  horizontal  sections  shall  be  proportional  to 
the  strains  to  which  they  are  subjected  —  the  weight  of  the  bar 
being  considered. 

Let  <J  —  weight  of  a^  unit  of  volume, 

"W"  =  weight  of  the  whole  bar, 

K0  =  ?i=  the  section  at  B  (Eq.  (11)  ), 

Kx   =  the  upper  section, 
K    =  variable  section,  and 
x     =  variable  distance  from  B  upwards. 
Also  let  the  sections  be  similar  : 

Then  P  =  Pt  +  $  /*K  dx=  strain  on  any  section, 
as  D  C.    But  TK  is  the  ability  to  resist  this  strain  ; 

/.  P,  -f  ^  fKdx  =  TK.    Differentiate  this  and  we  have 
*  K  dx=  TcIK 
or  ~r  dx  =  -W  which  by  integrating  gives 

I 

Tpc  =  Nap.logK-f  C    ..... 

*  Am.  R.  K.  Times  (Boston),  Vol.  20,  p.  206. 


26 


THE   RESISTANCE    OF   MATERIALS. 


But  for  x  =  0,  we  have  K  =  K0  .*.  C  =  —  Nap.  log  K0  =  — 

T)  »  -f7- 

Nap.  log  *LL.   Hence,  Eq.  (12a)  becomes—  x  =  Nap.  log  &•    or, 

k**tjw»j*,  w+K*  •  tft*P~t*~?f^o  =s  t^Wt  ^^ 

passing  to  exponetials,  gives  eT     =  -==- 


V     P  { 

- 


(13) 


For  the  upper  section  K  =  Kj  and  a?  =  ^  .*.  Kt  =    £_16T    -     (14) 


W  equals, 


/  /" 

5    /    Kotf=J    / 

V      7 


p       J 


-1)-     -     (15) 


Example.  What  must  be  the  upper  section  of  a  wrought-iron  shaft  of  uni- 
form resistance  1,000  ft.  long,  so  that  it  will  safely  sustain  its  own  weight  and 
75,000  Ibs. 

Let  T  =  10,000  Ibs.,  and 

$  =  0.27  Ibs.  per  cubic  inch. 

Then  Eq.  (11)  gives  K0  =  7.5  sq.  inches,  and 

equation  (14)  gives  Kx  =  10.37  inches. 

In  these  formulas  the  form  of  section  does  not  appear.  For 
tensile  strains,  the  strength  is  practically  independent  of  the  form, 
but  not  so  for  compression.  When  it  yields  by  crushing,  the 
influence  of  form  is  quite  perceptible,  but  not  so  much  so  as 
when  it  yields  by  bending  under  a  compressive  strain.  The. 
latter  case  will  be  considered  under  the  head  of  flexure. 


FIG.  11. 


STRAINS    IN    A    CLOSED    CYLINDER. 

If  a  closed  cylinder  is  subjected  to 
an  internal  pressure,  it  will  tend  to 
burst  it  by  tearing  it  open  along  a 
rectilineal  element,  or  by  forcing  the 
head  oft'  from  the  cylinder,  by  rup- 
turing it  around  the  cylinder.  First, 
consider  the  latter  case.  The  force 
which  tends  to  force  the  head  off  is 
the  total  pressure  upon  the  head,  and 
the  resisting  section  is  the  cylindrical 
annulus. 


TENSION.  27 

Let  D  =  the  external  diameter, 
d  —  the  internal  diameter, 
p  =  the  pressure  per  square  inch,  and 
t  =  the  thickness  of  the  cylinder. 
Then  ^d"p—tliG  pressure  upon  the  head  ; 

3—  d*)=tlie  area  of  the  cylindrical  armulus  ; 
2—  cF')=the    resistance    of    the    annulus  ;    and, 


O  :ft  -       . 

(16) 
which  solved  gives  t  =  (—  1  +  v/l  +•?  Y?  •  (17) 

Next  consider  the  resistance  to  longitudinal  rupturing.  As 
it  is  equally  liable  to  rupture  along  any  rectilinear  element, 
suppose  that  the  cylinder  is  divided  by  any  plane  which  passes 
through  the  axis.  The  normal  pressure  upon  this  plane  is  the 
force  which  tends  to  rupture  it,  and  for  a  unit  of  length  is 

pd 

and  the  resisting  force  is 


hence,  for  equilibrium, 

2Tt=pd    -  (18) 

The  value  of  t  from  (18)  divided  by  that  of  t  from  (16)  gives 

the  ratio  —  j  —  >  and  since  D  always  exceeds  d,  this  ratio  is  greater- 

than  2  ;  hence  there  is  more  than  twice  the  danger  of  bursting 
a  boiler  longitudinally  that  there  is  of  bursting  it  around  an 
annulus  when  the  material  is  equally  strong  in  both  directions. 
The  last  equation  was  established  by  supposing  that  all  the 
cylindrical  elements  resisted  equally,  but  in  practice  they  do 
not  ;  for,  on  account  of  the  elasticity  of  the  material,  they  will 
be  compressed  in  the  direction  of  the  radius,  thus  enlarging  the 
internal  diameter  more  than  the  external,  and  causing  a  corre- 
sponding increase  of  the  tangential  stress  on  the  inner  over  the 
outer  elements.  In  a  thick  cylindrical  annulus  it  is  necessary 
to  consider  this  modification. 

A  ofrL  -  2  TYM-t  if*  dh  : 


28  THE   RESISTANCE   OF    MATERIALS. 

To  find  the  VARYING  LAW  OF  TANGENTIAL  STRAINS,  let  D  and  d 
be  the  external  and  internal  diameters  before  pressure,  and 
D-j-2  and  d+y  the  corresponding  diameters  after  pressure. 
Then,  as  a  first  approximation  —  which  is  near  enough  for  prac- 
tice —  suppose  that  the  volume  of  the  annulus  is  not  changed, 
and  we  have 


or,  ~Dz=dy  nearly     ........     (19) 

But  the  strain  upon  a  cylindrical  filament  varies  as  its  elon- 
gation divided  by  its  length;  see  equation  (3).  Hence  the 
strain  on  the  external  annulus,  compared  with  the  internal, 
is  as 


z       y 

~D~  ~^d~  ~  or  as  D  to  d 

which  combined  with  (19)  gives 

d        1 
jj3  to  ~f  or  as  d*  to  D',  or  as  r*  to  R* 

where  r  and  R  are  radii  of  the  annulus. 

Hence,  the  strain  varies  inversely  as  the  square  of  the  dis- 
tance from  the  axis  of  the  cylinder. 

To    FIND   THE   TOTAL    RESISTANCE,  let 

x  =  the  variable  distance  from  the  axis  of  the  cylinder, 

T  =  the  modulus  of  rupture,  or  of  strain,  and 

t  =  the  thickness  of  the  annulus. 

Then  Tdx  is  the  strain  on  an  element  at  a  distance  r  from 
the  axis  of  the  cylinder,  or  otherwise  upon  the  inner  surface  of 
the  cylinder  ;  and  according  to  the  principle  above  stated, 

ra 
T—  a  dx  is  the  strain  on  any  element,  and  the  total  strain  on  both 

sides  is 

,R 


/f\ 
^=2 
. 


2Tr'     /    ^=2T— ^ 

t 

r 

If  t  —  r.  this  becomes 

Tt 

which  compared  with  equation  (18)  shows  that  when  the  thick- 
ness equals  the  radius,  the  resistance  is  only  half  what  it  would 


TENSION.  29 

be  if  the  material  were  non-elastic.  In  (20)  if  t  is  small  com- 
pared with  r,  it  becomes  2Ttf  nearly,  which  is  the  same  as 
equation  (18). 

If  the  ends  of  the  cylinder  are  capped  with  hemispheres,  the 
stress  upon  an  elementary  annulus  at  the  inner  surface  is 
%*Trdx.*  Proceeding  as  before,  and  we  find  that  the  total 
stress  necessary  to  force  the  hemispherical  heads  off  is 


which  is  also  the  stress  necessary  to  force  asunder  a  sphere  by 
internal  pressure,  when  the  elasticity  is  considered. 

If  cylinders  are  formed  by  riveting  together  plates  of  iron, 
their  strength  will  be  much  impaired  along  the  riveted  section. 
The  condition  of  the  riveted  joint  w^ill  doubtless  have  much 
more  to  do  .with  the  strength  than  the  compressibility  of  the 
material,  and  will  hereafter  be  considered. 


*  T.  J.  Rodman  says  the  resistance  on  any  elementary  annulus  is  T2nxdx 
(Exp.  on  metal  for  cannon,  p.  44) ;  but  it  appears  to  me  that,  to  make  his 
expression  correct,  T  must  be  the  modulus  at  any  element  considered,  and 
hence  variable,  whereas  it  should  be  constant.  The  strain  on  any  elementary 

r2 
annulus  whose  distance  is  x  from  the  centre  of  the  sphere,  is  T2nrdx,  —5  — 

dx  X 

2*r3T  — ;  and  the  total  resistance  is  the  integral  of  this  expression  between 
x 

the  limits  of  r  and  r+t. 


30 


THE    RESISTANCE    OF   MATERIALS. 


RESISTANCE    OF   GLASS  GLOBES    TO    INTERNAL    PRES- 
SURE. 


EXPERIMENTS    OF    WM.    FAIRBAIRN. 


Description  of  the  glass. 

Diameter  in 
inches. 

Thickness  in 
inches. 

Bursting  pres- 
sure in  Ibs.  per 
square  inch. 

Bursting  pres- 
sure in  Ibs.  per 
square  inch  of 
section. 

Flint-glass 

4.0  x  3.98 
40  x  3.98 

0.024 

0  025 

84 
93 

3504 
3720 

4 
4.5  x  4.55 
6 

0.038 
0.056 
0.059 

150 
280 
152 

3947 

5(525 
3864 

.  .  .  .4132 

Green-glass                .  . 

4.95 
495 

x 

Y 

5.0 
50 

0.022 
0.020 

90 

85 

5113 
5312 

4.0 
4.0 

X 
X 

4.05 
4.03 

0.018 
0.016 

84 
82 

4666 
5126 

Mean 

5054 

Crown-glass  

4.2 
4.05 

X 
Y 

4.35 

49, 

0.025 
0.021 

120 

126 

5040 
6000 

5.9 
6.0 

X 
X 

5.8 
6.3 

0.016 
0.020 

69 

86 

6350 
6450 

Mean.. 


.5960] 


The  following  table  exhibits  the  tensile  strength  of  cylindrical 
glass  bars  according  to  the  experiments  of  Fairbairn : — 


Description  of  the  glass. 

Area  of  specimen  in 
inches. 

Breaking  weight  in 
Ibs. 

Tenacity  per  square 
inch. 

Annealed  flint-glass.  .  . 

(  0.255 
1  0.196 

583 
254 

2286 
2540 

Green  -glass        

0.220 

639 

2896 

0.229 

583 

2546 

TENSION. 


31 


As  might  have  been  anticipated,  the  tenacity  of  bars  is  much 
less  than  globes  ;  for  it  is  difficult  to  make  a  longitudinal  strain 
without  causing  a  transverse  strain,  and  the  latter  would  have  a 
very  serious  effect ;  it  is  also  probable  that  the  outer  portion  of 
the  annealed  glass  is  stronger  than  the  inner,  and  there  is  a 
larger  amount  of  surface  compared  with  the  section,  in  globes 
than  in  cylinders. 


RIVETED     PLATES. 

RIVETED  PIRATES  are  used  in  the  construction  of  boil- 
ers, roofs,  bridges,  ships,  and  other  frames.  It  is  desirable  to 
know  the  best  conditions  for  riveting,  and  the  strength  of  riveted 
plates  compared  with  the  solid  section  of  the  same  plates.  The 
common  way  of  riveting  is  to  punch  holes  through  both  plates, 
into  which  red-hot  bolts  or  rivets  are  placed,  and  headed  down 
while  hot.  The  process  of  punching  strains,  and  hence  weakens, 
the  material.  A  better  way  is  to  bore  the  holes  in  the  plates, 
and  then  rivet  as  before. 
The  holes  in  the  separate 
plates  should  be  exactly 
opposite  each  other,  so  that 
there  will  be  no  side  strain 
on  the  plates  caused  by 
driving  the  rivets  home, 
and  to  secure  the  best  ef- 
fects of  the  rivets  them- 
selves. They  are  some- 
times placed  in  single  and 
sometimes  in  double  rows, 
and  experiment  shows  that  the  latter  possesses  great  advantage 
over  the  former.  Experiments  have  been  made  upon  plates  of 
the  form  shown  in  Fig.  12,  both  with  lap  and  butt-joints,  and 
with  single  and  double  rows  of  rivets.* 

*  Lond.  Phil.  Transactions,  part  3d,  1850,  p.  677. 


FIG.  12. 


<. 


32 


THE    RESISTANCE    OF   MATERIALS. 


Table  sJwwing  the  strength  of  single  and  double  riveted  plates. 


Cohesive  strength  of  the  plates 
in  Ibs.  per  square  inch. 
T. 

Strength  of  single-riveted  joints 
of  equal  section  to  the  plates, 
taken  through  the  line  of  riv- 
ets'.    Breaking  weight  in  Ibs. 
per  square  inch. 

Strength  of  double-riveted  joints 
of  equal  section  to  the  plates, 
taken  through  the  line  of  riv- 
ets.    Breaking  weight  in  Ibs. 
per  square  inch. 

57,724 
61,579 
58,322 
50,983 
51,130 
49,281 

45,743 
36,606 
43,141 
43,515 
40,249 
44,715 

52,352 

48,821 
58,286 
54,594 
53,879 
53,879 

Mean.  .54,836 

42,328 

53,635 

It  will  be  observed  that  in  double-riveting  there  is  but  little 
loss  of  strength,  while  there  is  considerable  loss  in  single-rivet- 
ing. In  the  preceding  experiments  the  solid  section  of  the 
plates,  taken  through  the  centre  of  the  rivet-holes,  was  used ; 
but,  as  Fairbairn  justly  remarks,  we  must  deduct  30  per  cent, 
for  metal  actually  punched  out  to  receive  the  rivets.  But  as 
only  a  few  rivets  came  within  the  limits  of  the  experiments, 
and  as  an  extensive  combination  of  rivets  must  resist  more 
effectually,  and  as  something  will  be  gained  by  the  friction 
between  the  plates,  it  seems  evident  that  we  may  use  more  than 
60  per  cent,  of  the  strength  of  riveted  plates  as  indicated  above. 
Fairbairn  says  we  may  use  the  following  proportions : — 

Strength  of  plates 100 

Strength  of  double -rive  ted  plates 70 

Strength  of  "single -riveted  plates 56 

Size  and  distribution  of  rivets. — The  best  size  of  the  rivets, 
the  distance  between  them,  and  the  proper  amount  of  lap  of  the 
plates,  can  be  determined  only  by  long  experience,  aided  by 
experiments.  Fairbairn  gives  the  following  table  as  the  results 
of  his  information  upon  this  important  subject,  to  make  the 
joint  steam  or  water  tight : — 


TENSION. 


33 


Table  showing  the  strongest  forms  and  best  proportions  of  riveted  joints,  as  deduced 
from  experiments  and  actual  practice.  ( Useful  Information  for  Engineers, 
1st  Series,  p.  285.) 


Thickness  of 
plates  in 
inches, 
t. 

Diameter  of 
the  rivets  in 
inches, 
d. 

Length  of 
rivets  from  the 
head  in  inches. 
1. 

Distance  of 
rivets  from  cen- 
tre to  centre  in 
inches, 
a. 

Quantity  of 
lap  in  single 
joints  in  inches. 
b. 

Quantity  of 
lap  in  double- 
riveted  joints 
in  inches. 
c. 

A*  to  A 

2   t 

ttt 

6t 

6t 

lot 

A 

a 

u 

5t 

u 

u 

A 

n 

u 

a 

tiki 

8it 

A  toil 

1*4 

u 

4t 

4t 

61  t 

STRENGTH    OF     IRON   IN    DIFFERENT    DIRECTIONS    OF 
THE    ROLLED   SHEET.* 

In  obtaining  specimens  for  these  experiments,  care  was  gen- 
erally taken  to  have  them  cut  in  different  directions  of  the  roll- 
ing, longitudinally  and  transversely,  and  in  some  cases  diag- 
onally, to  that  direction.  The  table  will  be  found  to  indicate 
the  direction  of  slitting  in  each  case,  and  the  comparison  con- 
tained in  the  table  is  given  to  show  what  information  the  in- 
quiry has  elicited. 

The  comparison  is  made  principally  on  the  minimum  strength 
of  each  bar,  being  that  which  can  alone  be  relied  on  in  practice  ; 
for  if  the  strength  of  the  weakest  point  in  a  boiler  be  overcome, 
it  is  obviously  unimportant  to  know  that  other  parts  had  greater 
strength.  In  one  case,  however,  two  bars,  one  cut  across  the 
direction  of  rolling,  and  the  other  longitudinally,  were,  after  be- 
ing reduced  to  uniform  size,  broken  up  cold,  with  a  view  to  this 
question.  The  result  showed  that  the  length-strip  was  TTV  per 
cent,  stronger  than  the  one  cut  crosswise,  considering  the  tenacity 
of  the  latter  equal  to  100.  Of  the  other  sets,  embracing  about 
40  strips  cut  in  each  direction,  it  appears  that'  some  kinds  of 
boiler  iron  manifest  much  greater  inequality  in  the  two  direc- 
tions than  others.  It  is  in  certain  cases  not  much  over  one  per 
cent.,  and  in  others  exceeds  twenty,  and  as  a  mean  of  the  whole 
series  it  may  be  stated  to  amount  to  six  per  cent,  of  the  strength 
of  the  cross-cut  bars.  The  number  of  trials  on  those  cut  diag- 
onally is  not  perhaps  sufficiently  great  to  warrant  a  general  de- 
duction ;  but,  so  far  as  they  go,  they  certainly  indicate  that  the 
strength  in  this  direction  is  less  than  either  of  the  others. 


3 


*  Experiments  of  Franklin  Institute. 


34  THE   RESISTANCE   OF   MATERIALS. 

Had  we  compared  the  mean  instead  of  the  least  strength  of  bars 
as  given  in  the  table,  the  result  would  not  have  differed  materi- 
ally in  regard  to  the  relative  strength  in  the  respective  directions. 

The  boiler-iron  manufactured  by  Messrs.  E.  H.  &  P.  Ellicott, 
which  was  tried  in  all  these  modes  of  preparation  of  specimens, 
gave  the  following  results : 

1.  When  tried  at   original  sections,  seven   experiments  on 
length-sheet  specimens  gave  a  mean  strength  of  55285  Ibs.  per 
square    inch,  the   lowest   being   44399   Ibs.,  and  the  highest 
59307  Ibs.    Fourteen  experiments  on  cross-sheet  specimens  gave 
a  mean  of  53896  Ibs.,  the  lowest  result  being  50212  Ibs.,  the 
highest  58839  Ibs. ;  and  six  experiments  on  strips  cut  diagonally 
from  the  sheet  exhibited  a  strength  of  53850  Ibs.,  of  which  the 
lowest  was  51134  Ibs.,  and  the  highest  58773  Ibs. 

2.  When  tried  by  filing  notches  on  the  edges  of  the  strips,  to 
remove  the  weakening  effect  of  the  shears,  the  length-sheet  bars 
gave,  at  fourteen  fractures,  a  mean  strength  of  G3946  Ibs.,  vary- 
ing between  56346  Ibs.  and  78000  Ibs.  per  square  inch.     The 
cross-sheet   specimens  tried    after  this   mode   of    preparation 
exhibited,  at  three  trials,  a  mean  strength  of  60236  Ibs.,  vary- 
ing from  55222  Ibs.  to  65143  Ibs. ;  and  the  diagonal  strips,  at 
four  trials,  gave  a  mean 'result  of  53925  Ibs.,  the  greatest  differ- 
ence being  between  51428  Ibs.  and  56632  Ibs.  per  square  inch. 

3.  Of  strips  reduced  to  uniform  size  by  filing,  four  compara- 
ble experiments  on  those  cut  lengthwise  of  the  sheet  gave  a  mean 
strength  of  63947  Ibs.,  of  which  the  highest  was  67378  Ibs.,  and 
the  lowest  60594  Ibs. 

Cross-sheet  specimens,  tried  after  the  same  preparation,  ex- 
hibited, at  thirty-three  fractures,  a  mean  of  50176  Ibs.,  of  which 
the  highest  was  65785  Ibs.,  and  the  lowest  52778  Ibs.  No  bar 
cut  diagonally  was  reduced  to  uniform  size. 

From  the  foregoing  statements  it  appears  that  by  filing  in 
notches  and  filing  to  uniformity,  we  obtain  results  63946  Ibs. 
and  63947  Ibs.  for  the  strength  of  strips  cut  lengthwise,  differing 
from  each  other  by  only  a  single  pound  to  the  square  inch,  and 
that  by  these  two  modes  of  preparation  the  cross-sheet  speci- 
mens gave  respectively  60236  Ibs.  and  60176  Ibs.,  differing  by 
only  60  Ibs.  to  the  square  inch.  This  seems  to  prove  that  by 
both  methods  of  preparing  the  specimens  the  accidental  weak- 
ening effect  of  slitting  had  been  removed  by  separating  all  that 


TENSION. 


35 


portion  of  the  metal  on  which  it  had  been  exerted.  Hence  we 
may  infer  that  the  differences  between  length-sheet  and  cross- 
sheet  specimens  are  really  and  truly  ascribable  to  a  difference  of 
texture  in  the  two  directions,  which  will  be  seen  to  amount,  in 
the  case  of  filing  in  notches,  to  6.15  per  cent.,  and  in  that  of 
filing  to  uniformity,  to  6.26  per  cent,  of  strength  of  cross-sheet 
specimens. 

Table  of  the  comparative  mew  of  the  strength  of  specimens  of  ten  different 
sorts  of  boiler  and  one  of  bar  iron,  in  the  longitudinal,  transverse,  and  diago- 
nal direction  of  the  rotting,  as  deduced  from  tJie  least  strength  of  each  specimen, 
and  the  average  minimum  of  each,  sort  of  iron,  in  eadi  direction  in  which  it  was 
tried. 


j 

1 

I 

I 

1 

1 

** 

1 

!§ 

1  • 

I 

1 

'-3  . 

Is 

I] 

If 

.[3 

ll 

ll 

|| 

.s'-s 

sS 

01 

at 

a  g 

•3 

f! 

I 

*o 

£•2 
tort 

1?*^ 

1 

6 

1 

i 

0 

E 

1 

1 

K 

8 

02 

H 

02 

02 

02 

2 

58977 

125 

57182 

Tilted. 

3 

53828 

130 

Tilted. 

57789 

4 

47167 

133 

do. 

53176 

6 

52280 

135 

do. 

47738 

8 

50103 

137 

do. 

50358 

Mean 

53324 

51191 

Mean 

57182 

55882 

49048 

42 

51653 

Puddled. 

142 

44399 

43 

44102 

do. 

143 

63136 

44 

58836 

do. 

146 

60594 

46 

59262 

H'd  pla.* 

148 

52468 

48 

59418 

do. 

149 

52228 

49 

57565 

do. 

150 

56869 

51 

ITd  pi. 

59656 

151 

53811 

53 

H'd  pla. 

56062 

152 

56073 

Mi 

Puddled. 

57926 

154 

51134 

58 

do. 

50570 

157 

52102 

59 

48308 

Puddled. 

160 

53862 

60 

58684 

do. 

162 

50212 

61 

52869 

do. 

164 

56346 

62 

57612 

do. 

167 

66682 

64 

Puddled. 

45392 

169 

54361 

65 

do. 

51255 

171 

55612 

68 

57929 

H'd  pla. 

174 

51425 

70 

47638 

do. 

71 

H'd  pla. 

64684 

Mean 

54253 

53646 

52568 

73 

do. 

52657 

74 

do'. 

49351 

Mean 

54074 

63049 

226 

227 

49053 
53699 

228 

40643 

229 

46473 

230 

49368 

Mean 

49368 

47467 

Hammered  and  rolled  into  plates. 


36 


THE   BESISTANCE   OF  MATEKIALS. 


The  specimens  from  42  to  74  were  partly  puddled  iron,  and 
partly  Juniata  blooms,  hammered  and  rolled  into  plate.  The 
length  and  cross-sheet  specimens  of  these  two  kinds  must  be 
compared  separately. 

All  the  experiments  on  No.  228  (cross)  and  230  (length)  were 
made  at  ordinary  temperatures  with  a  view  to  this  comparison. 

29.    TENSILE   STRENGTH   OF   WROUGHT   IRON  AT  VARIOUS 
TEMPERATURES. 

Mr.  Fairbairn  has  made  experiments  upon  rolled  plates  of 
iron,  and  rods  oijtrivet  iron,  at  various  temperatures.  The  for- 
mer were  broken  in  the  direction  of  the  fibre  and  across  it. 
The  specimen  when  subjected  to  experiment  was  surround- 
ed with  a  vessel  into  which  freezing  mixtures  were  placed 
to  produce  the  lower  temperatures,  and  oil  heated  by  a  fire 
underneath  to  produce  the  high  temperatures.  The  experi- 
ments were  made  upon  Staffordshire  plates,  which  are  inferior 
to  several  other  kinds  in  common  use.  The  following  table 
gives  a  summary  of  the  results : — 

Table  showing  the  Resistance  of  Staffordshire  Plates  at  Different  Temperatures. 


^ 

.a 

*- 

"2 

1 

1 

1 

&o  g 

SI 

I 

i 

li 

1| 

ft 

08  M 

Remarks, 

•g 

o 

m 

O  ,Q 

P 

"S  5 

1 

!& 

if  J 

tl 

H 

« 

PQ 

m  aS 

JMLI 

1 

0° 

0.6868 

33,660 

49,009 

49,009 

With. 

2 

60 

0.7825 

31,980 

40,357 

Across. 

3 

60 

0.6400 

27,780 

43,406 

•  44,498 

Across. 

4 

60 

0.6368 

31,980 

50,219 

With. 

5 

110 

0.6633 

29,460 

44,160 

Across.* 

6 

112 

0.6800 

28,620 

42,088 

-42,291 

With. 

7 

120 

0.8128 

37,020 

40,625 

With. 

8 

212 

0.8008 

31,980 

39,935 

With. 

9 

212 

0.6633 

30,300 

45,680 

45,005 

Across. 

10 

212 

0.6800 

33,660 

49,500 

With. 

11 

270 

0.6432 

28,620 

44,020 

44,020 

With. 

12 
13 

340 
340 

0.6400 
0.6800 

31,980 
28,620 

49,968 

42,088 

i  46,018 

With.f 
Across. 

14 

395 

0.6666 

30,720 

46,086 

46,086 

With. 

15 

16 

Scarcely  red 
Dull  red 

0.6200 
0.6076 

23,520 
18,540 

38,032 
30,513 

I  34,272 

Across. 
Across  4 

*  Too  high ;  fracture  very  uneven. 

f  Too  low ;  tore  through  the  eye. 

%  Too  high ;  the  specimen  broke  with  the  first  strain. 


TENSION. 


37 


The  mean  values  given  in  the  sixth  column  of  this  Table 
exhibit  a  remarkable  degree  of  uniformity  in  strength  for  all 
temperatures,  from  60  degrees  to  395  degrees.  The  single  ex- 
ample at  0  degrees  gives  a  higher  value  than  the  mean  of  the 
others,  but  not  higher  than  for  some  of  the  specimens  at 
higher  temperatures.  At  red  heat  the  iron  is  very  much 
weakened.  This  fact  should  be  noticed  in  determining  the 
strength  of  boiler-flues,  as  they  are  often  subjected  to  in- 
tense heat  when  not  covered  with  water. 

The  experiments  upon  rivet  iron  were  made  with  the  same 
machine,  and  in  the  same  manner,  the  results  of  which  are 
shown  in  the  following  table : — 

Table  showing  the  Results  of  Experiments  on  Rivet  Iron  at  Different  Tem- 
peratures. 


1 

.s 

|f 

n 

a 

1 

1 

|.S 

IS 

& 

jj 

1 

la 

•I 

*t| 

Remarks.       , 

•s 

I 

1 

1 

III 

iff 

17 
18 
19 

-30° 
-f60 
60 

0.2485 
0.2485 

15,715 
15,400 
15  820 

63,239 
61,971 
63  661 

63,239 

62,816 

Too  low. 
Too  low. 
Too  low 

20 

114 

17,605 

70,845 

70,845 

21 

212 

20,545 

82,676 

22 

212 

0.1963 

14,560 

74,153 

79,271 

23 

212 

0.2485 

20,125 

80,985 

24 
25 

250 
270 

0.1963 

0.2485 

16,135 
20,650 

82,174 
83,098 

82,636 

26 

27 

310 
325 

0.1963 
0.1963 

15,820 

17,185 

80,570 

87,522 

84,046 

28 
29 

415 
435 

0.2485 

20,335 
21,385 

81,830 
86,056 

I  83,943 

30 

Red  heat. 

8,965 

36,076 

35,000 

Too  high 

From  this  Table  we  see  that  there  is  a  gradual  increase  of 
strength  from  60  degrees  to  325,  where  it  appears  to  attain  its 
maximum.  The  increase  is  a  very  important  amount,  being 
about  30  per  cent. 

It  is  a  little  remarkable  that  the  specimen  at  minus  30  de- 
grees is  stronger  than  the  mean  of  the  two  at  60  degrees ;  but 
we  observe,  as  before,  that  it  is  not  as  strong  as  some  of  the 
single  specimens  at  higher  degrees. 


38 


THE   RESISTANCE    OF   MATERIALS. 


Mr.  Johnson,  when  in  the  employ  of  the  Navy  Department, 
in  1844,  made  some  experiments  to  determine  the  effects  of 
thermo-tension  upon  different  kinds  of  iron.*  He  took  two 
bars  of  the  same  kind  of  iron,  and  of  the  same  size,  and  broke 
one  while  cold.  He  then  subjected  the  other  to  the  same  ten- 
sion when  heated  400  degrees,  after  which  the  strain  was  re- 
lieved, and  the  bar  was  allowed  to  cool,  and  the  permanent 
elongation  noted,  after  wiiich  it  was  broken  by  an  additional 
load.  It  will  thus  be  seen  that  the  experiments  were  not  con- 
ducted in  the  same  way  as  those  by  Fairbairn.  The  following 
table  gives  the  results  of  his  experiments  : 

The  Results  of  Experiments  an  Thermo- Tewion,  at  400°  Temperature. 


o>  5 

mi 

. 

j 

•a 

£  (- 

"So-S  c  * 

H 

C 

l| 

KIND  OF  IRON. 

8 
ja 

*|| 

*ll  % 

1 

i 

"&T3   S3 

c 

*O  S   rt  J2   ° 

us 

tac  >• 

00 

ill 

ill 

1 

ill  I 

Sltal 

if 

Tons. 

Tons. 

Inches. 

Per  cent. 

Per  cent. 

Per  cent. 

Tredegar,  round.  .  . 

60 

71.4 

1.91 

6.51 

19.00 

25.51 

Tredegar,  round  .  .  . 

60 

72.0 

1.91 

(6.51) 

20.00 

26.51 

Tredegar,  square  bar 

60 

67.2 

1.69 

6.77 

12.00 

18.77 

Tredegar,  r'nd,  No.  3 

58 

68.4 

1.15 

5.263 

17.93 

23.19 

Salisbury,  round  .  .  . 

105.87 

121.0 

3.59 

3.73 

14.64 

18.37 

M 

ean  

5.75 

16.64 

22.40 

Remarks. — From  the  two  former  sets  of  experiments,  pp. 
36  and  37,  it  appears  that  the  strength  of  the  iron  was  in- 
creased by  an  increase  of  temperature  at  the  time  the  bar 
was  broken,  and  by  the  latter  that  it  was  not  only  increased,  but, 
by  being  subjected  to  severe  tension  while  at  a  high  temper- 
ature, the  increased  strength  was  not  lost  by  cooling.  It  hardly 
seems  probable  that  this  increased  strength  would  be  retained 
indefinitely,  and  hence  it  is  important  to  know  how  long  it  was 
after  the  piece  was  cooled  before  it  was  broken. 

These  results  are  confirmed  by  the  experiments  of  the  com- 
mittee of  the  Franklin  Institute,  as  shown  by  the  following 
table.  See  Journal  of  the  Franklin  Institute,  vol.  xx.,  3d  series, 
p.  22. 


*  Senate  Doc.,  No.  1,  28th  Cong.,  3d  Sess.,  1844-5,  p. 


TENSION. 


39 


ABSTRACT   OF  A  TABLE 

Of  the  c&mparatlve  view  of  the  influence  of  higJi  temperatures  on  the  strength  of 
iron,  as  exhibited  by  73  experiments  on  47  different  specimens  of  that  metal, 
at  46  different  temperatures,  from  212°  to  1317°  Fahr.,  compared  with  tlie 
strength  of  each  bar  ichen  tried  at  ordinary  temperatures,  the  number  of  expe- 
riments at  the  latter  being  163. 


No.  of  the  experi- 
ment. 

Temperature  observed 
at  moment  of  fracture. 

Strength  at  ordinary 
temperature. 

Strength  at  the  tem- 
perature observed. 

1 

212° 

56736 

67939 

2 

214 

53176 

61161 

3 

394 

68356 

71896 

9 

440 

49782 

59085 

10 

520 

54934 

58451 

15 

554 

54372 

61680 

20 

568 

67211 

76763 

25 

574 

76071 

65387 

40 

722 

57133 

54441 

45 

824 

59219 

55892 

50 

1037 

58992 

37764 

58 

1245 

54758 

20703 

59 

1317 

54758 

18913 

Remark. — According  to  these  experiments,  as  shown  in  the 
4th  column,  the  strength  increases  with  the  temperature  to  394 
degrees,  when  it  attains  its  maximum  ;  although  in  some  cases 
the  strength  was  increased  by  increasing  the  temperature  to  568 
degrees.  By  comparing  the  3d  and  4th  columns  we  see  that 
the  strength  is  greater  for  all  degrees  from  212°  to  574°  than 
it  is  at  ordinary  temperatures,  but  above  574°  it  is  weaker. 
The  experiments  on  Salisbury  iron  showed  that  the  maximum 
tenacity  was  15.17  per  cent,  greater  than  their  mean  strength 
when  tried  cold.  The  committee  above  referred  to  determined 
the  maximum  strength  of  about  half  the  specimens  used  in  the 
preceding  table  by  actual  experiment,  and  calculated  it  for  the 
others ;  and  from  the  results  derived  the  following  empirical 
formula  for  the  diminution  in  strength  below  the  maximum  for 
high  degrees  of  heat : — 

D5  =  c(6  -  SO)13 

in  which  D  is  the  diminution  after  it  has  passed  the  maximum, 
0  the  temperature  Fahrenheit,  and 
c  a  constant. 


40  THE   RESISTANCE   OF   MATERIALS. 

This  formula  appears  to  be  sufficiently  exact  for  all  tem- 
peratures between  520°  and  1317°. 

3O.    EFFECT  OF  SEVERE  STRAINS    UPON    THE    ULTIMATE 

TENACITY  OF  IRON  RODS. — Thomas  Loyd,  Esq.,  of  England, 
took  20  pieces  of  If  S.  C.  ||j>  bar  iron,  each  10  feet  long,  which 
were  cut  from  the  middle  of  as  many  rods.  Each  piece  was  cut 
into  two  parts  of  5  feet  each,  and  marked  with  the  same  letter. 
A,  B,  C,  &c.,  were  first  broken,  so  as  to  get  the  average  breaking 
strain.  A2,  B2,  &c.,  were  subjected  to  the  constant  action  of 
three-fourths  the  breaking  weight,  previously  found,  for  five 
minutes.  The  load  was  then  removed,  and  the  rods  afterwards 
broken. 

Results  of  the  Experiments.  * 


FIRST. 

SECOND. 

Mark  on  the  bars. 

Breaking  weight  in 
tons  (gross). 

Mark. 

Breaking  weight  in    ; 
tons. 

A 

33.75 

A  2 

33.75 

B 

30.00 

B  2 

33.00 

C 

33.25 

C  2 

33.25 

D 

32.75 

D2 

32.25 

E 

32.50 

E  2 

32.50 

F 

33.25 

F  2 

33.00 

G 

32.75 

G2 

33.00 

H 

33.25 

H2 

33.50 

I 

33.50 

I  2 

32.75 

J 

33.50 

J  2 

33.25 

K 

32.25 

K2 

32.50 

L 

32.25 

L  2 

31.50 

M 

30.25 

M2 

32.75 

N 

34.25 

N2 

34.00 

0 

31.75 

0  2 

32.50 

P 

29.75 

P  2 

31.00 

Q 

33.50 

Q  2 

33.75 

B 

33.75 

R  2 

33.75 

S 

33.00 

S  2 

33.25 

T 

32.25 

T  2 

31.00 

Mean 

32.57 

32.81 

We  here  see  that  a  strain  of  25  tons,  or  three-fourths  the 
breaking  weight,  did  not  weaken  the  bar. 

*  Fairbairn,  Useful  Information  for  Engineers,  First  Series,  p.  313. 


TENSION. 


41 


These  experiments  indicate  that  a  frame  or  bridge  may  be 
subjected  to  a  severe  strain  of  three-fourths  of  its  strength  for 
a  short  time  without  endangering  its  ultimate  strength. 

3  1 .    EFFECT  OF  REPEATED  RUPTURE. The  following  eX- 

periments  were  made  at  Woolwich  Dockyard,  England.  The 
same  bar  was  subj  ected  to  three  or  four  successive  ruptures  by 
tensile  strains.  They  show  the  remarkable  fact,  that  while  great 
strains  impair  the  elasticity,  as  shown  by  Hodgkinson,  yet  they 
do  not  appear  to  diminish  the  ultimate  tenacity.  This  fact  is 
important,  for  it  shows  that  iron,  which  has  been  broken  by 
tension  in  a  structure,  may  safely  be  used  again  for  any  strain 
less  than  that  for  which  it  was  broken. 

Table  stowing  the  effect  of  repeated  Fracture  on  Iron  Bars. 


First  br 

eakage. 

Second  t 

reakage 

Third  br 

eakage. 

Fourth  b 

reakage. 

Reduced 

from 

Mark. 

sectional 

Tons. 

Stretch 
in  54 
inches. 

Tons. 

Stretch 
in  36 
inches. 

Tons. 

Stretch 
hi  24 
inches. 

Tons. 

Stretch 
in  15 
inches. 

1.37  sqr. 
inches  to 

.- 

In. 

In. 

In. 

A. 

33.75 

0.9125 

35.50 

0.200 

B 

33.75 

0.9250 

35.25* 

0.225 

37.00 

1.00 

38.75 

1.25 

E 

32.50 

0.9250 

34.75 

0.125 

F 

33.25 

1.0500 

35.50 

0.112 

37.25 

0.62 

40.40 

1.18 

G 

32.75 

0.8500 

35.00 

0.125 

37.50 

40.41 

1.25 

H 

33.75 

1.0(525 

36.25 

0.187 

I 

33.50 

0.8375 

34.50 

0.62 

36.50 

1.50 

J 

33.50 

0.9250 

36.00 

0.025 

36.75 

1.12 

41.75 

1.25 

L 

32.25 

Defect'e 

36.50 

0.150 

37.75 

41.00 

0.31 

1.25 

M 

30.25 

Defect  'e 

36.50 

.62 

37.75 

0.60 

38.50 

0.06 

1.25 

Mean  

32.95 

35.57 

37.21 

40.16 

1.24 

Mean  pr.  sq.  in. 

24.04 

25.93 

27.06 

29.20 

0.90    ' 

We  thus  see  that  while  the  section  is  reduced  10  per  cent.,  the 
strength  is  apparently  increased  over  20  per  cent.  It  is  not, 
however,  safe  to  infer  that  the  strength  is  actually  increased, 
for  it  is  probable  that  it  broke  the  first  time  at  the  weakest 
point,  and  the  next  time  at  the  next  weakest  point,  and  so  on. 

We  also  observe  that  the  total  elongations  are  not  proportional 
to  the  tensile  strains,  which  is  in  accordance  with  the  results 
of  other  experiments. 

ANNEALING. 

32.  ANNEALING  is  a  process  of  treating  metals  so  as  to  make 


4:2  THE   RESISTANCE    OF   MATERIALS. 

them  more  ductile.  To  secure  this,  the  metals  are  subjected  to 
a  high  heat  and  then  allowed  to  cool  slowly.  Steel  is  softened 
in  this  way,  so  that  it  may  be  more  easily  worked.  Campin  * 
says  that  steel  should  not  be  overheated  for  this  purpose. 
Some  bury  the  heated  steel  in  lime ;  some  in  cast-iron  borings  ; 
and  some  in  saw-dust.  He  (Campin)  says  the  best  plan  is  to 
put  the  steel  into  an  iron  box  made  for  the  purpose,  and  fill  it 
with  dust-charcoal,  and  plug  the  ends  up  to  keep  the  air  from 
the  steel ;  then  put  the  box  and  its  contents  into  a  fire  until  it 
is  heated  thoroughly  through,  and  the  steel  to  a  low  red  heat. 
It  is  then  removed  from  the  fire,  and  the  steel  left  in  the  box 
until  it  is  cold.  Tools  made  of  annealed  steel  will,  in  some 
cases,  last  much  longer  than  those  made  of  unannealed  steel. 

But  it  appears  from  the  following  table  that  it  weakens  won 
to  anneal  it. 

Table  of  the  strength  of  WrougJit  Iron  Annealed  at  Different  Temperatures. 


No.  of  com- 
parisons. 

Strength  at  or- 
dinary   temp, 
before  anneal- 
ing. 

Temperature  at  which  an- 
nealing took  place. 

Strength  at 
the  annealing 
temperature. 

Strength  after 
annealing  and 
cooling. 

Ratio    of    di- 
minution    of 
strength. 

1 

5 

10 
15 
17 
18 
19 

57,133 

53,774 
52,040 
48,407 
73,880 
76,986 
89,162 

1037° 
1155 
1245 
Bright  welding  heat. 
Low  welding  heat. 
Bright  welding  heat. 
Low  welding  heat. 

*   37,764 
21,967 
20,703 

55,678 
45,597 
38,843 
38,676 
53,578 
50,074 
48,144 

0.025 
0.152 
.253 
.201 
.275 
.349 
.460 

33.    THE  STRENGTH  OF  IKON  AND  STEEL  AL.SO  DEPENDS 

largely  upon  the  processes  of  their  manufacture,  and  their 
treatment  afterwards.  The  strength  of  wrought  iron  de- 
pends upon  the  ore  of  which  it  is  made ;  the  manner  in 
which  it  is  smelted  and  puddled,  the  temperature  at  which 
it  is  hammered,  and  the  amount  of  hammering  which 
it  receives  in  bringing  it  into  shape.  The  same  remark 
applies  to  cast  steel.  If  the  former  is  hammered  when 
it  is  comparatively  cold,  it  will  weaken  it,  especially  if  the  blows 
are  heavy ;  but  the  latter,  steel,  may  be  greatly  damaged,  or 
even  rendered  worthless  by  excessive  heat,  and  it  is  greatly 
improved  by  hammering  when  comparatively  cold.  For  the 
effect  of  tempering  on  the  crushing  strength,  see  Article  50. 


Campin's  "  Practical  Mechanics,"  p.  364. 


TENSION.  43 

34.  PROLONGED  FUSION  OF  CAST  IRON. — Cast  iron  is  also 
subjected  to  great  modifications  of  strength  on  account  of 
the  manipulations  to  which  it  is  or  may  be  subjected  in  its 
manufacture  and  preparations  for  use.  The  strength  in  some 
cases  is  greatly  increased  by  keeping  the  metal  in  a  fused  state 
some  time  before  it  is  cast.  Major  Wade  made  experiments 
upon  several  kinds  of  iron,  all  of  which  were  increased  in 
strength  with  prolonged  fusion  (see  Rep.  p.  44),  one  example 
of  which  is  given  in  the  following 

Table  showing  the  Effects  of  Prolonged  Fusion. 


Tensile  Strength  in 
Ibs.  per  sq.  in. 

Iron  in  fusion     

^  hour 

17,843 

k  «          it            U 

1     " 

20,127 

u       u       u 

.    .           1^-  hour 

24,387 

u       u       u 

2   hours 

34,496 

3«l»  EFFECT  OF  REUIELTING  CAST  IRON. But  the  great- 
est effect  was  produced  by  remelting.  The  density,  tenacity, 
and  transverse  strength  were  all  increased  by  it,  within  certain 
limits.  For  instance,  a  specimen  of  !Nb.  1  Greenwood  pig- 
iron  gave  the  following  results  : — (Rep.,  p.  279.) 

Table  showing  the  Effects  of  Remelting. 


No.  1  Greenwood  iron. 

Density. 

Tensile  Strength. 

Crude  pig-iron             .               ....       

7.032 

14000 

"      remelted  once     

7.086 

20,900 

"            "         twice  

7.198 

30,229 

"            "          three  times                        

7  301 

35  786 

But  there  is  a  point  beyond  which  remeltings  will  wreaken  the 
iron.  Mr.  Fairbairn  made  an  experiment  in  which  the  strength 
of  the  iron  was  increased  for  twelve  remeltings,  and  then  the 
strength  decreased  to  the  eighteenth,  where  the  experiment  ter- 
minated. In  some  cases  no  improvement  is  made  by  remelting, 
but  the  iron  is  really  weakened  by  the  process ;  so  that  it  be- 
comes necessary  to  determine  the  character  of  each  iron  under 
the  various  conditions  by  actual  experiment. 

The  laws  which  govern  Greenwood  iron  were  so  thoroughly 


44  THE  RESISTANCE   OF   MATEEIALS. 

determined  that  the  results  which  will  follow  from  any  given 
course  of  treatment  may  be  predicted  with  much  certainty. 
(Rep.,  p.  245.) 

By  mixing  grades  Kos.  1,  2,  and  3,  and  subjecting  them  to  a 
third  fusion,  one  specimen  was  obtained  whose  density  was  7.304, 
and  whose  tenacity  was  45,970  pounds,  which  is  the  strongest 
specimen  of  cast  iron  ever  tested.  (Rep.,  p.  279.) 

As  a  general  result  of  these  experiments,  Major  Wade  re- 
marks (p.  243),  "  that  the  softest  kinds  of  iron  will  endure 
a  greater  number  of  meltings  with  advantage  than  the  higher 
grades.  It  appears  that  when  iron  is  in  its  best  condition  for 
casting  into  proof  bars  (that  is,  small  bars  for  testing  the  metal) 
of  small  bulk,  it  is  then  in  a  state  which  requires  an  additional 
fusion  to  bring  it  up  to  its  best  condition  for  casting  into  the 
massive  bulk  of  cannon." 

36.  THE  MANNER  OF  COOLING  also  affects  the  strength. 
It  was  found  that  the  tensile  strength  of  large  masses  was  in- 
creased by  slow   cooling  ;    while  that  of    small  pieces  was 
increased  by  rapid  cooling.     (Rep.,  p.  45.) 

37.  THE  MODULUS  OF  STRENGTH  IS  MODIFIED,  W6    thllS 

see,  by  a  great  variety  of  circumstances ;  and  hence  it  is  im- 
possible to  assign  any  arbitrary  value  to  it  for  any  material, 
that  will  be  both  safe  and  economical ;  but  its  value  must  be 
determined,  in  any  particular  case,  by  direct  experiment,  or 
something  in  regard  to  the  quality  of  the  material  must  be 
known  before  its  approximate  value  can  be  assumed. 

38.  SAFE  LIMIT  OF  LOADING. — Structures  should  not  be 
strained  so  severely  as  to  damage  their  elasticity.     According 

Article^9,  it  appears  that  a  weight  suddenly  applied  will 
produce  twice  the  elongation  that  it  will  if  applied  gradually 
or  by  increments.  Hence,  structures  which  are  subjected  to 
shocks  by  sudden  applications  of  the  load,  should  be  so  propor- 
tioned as  to  resist  more  than  double  the  load  as  a  constant 
dead-weight  without  straining  it  beyond  the  elastic  limit. 

This  method  of  indicating  the  limits,  suggested  by  M.  Pon- 
celet,  is  perfectly  rational ;  but,  unfortunately,  the  elastic  limits 
have  not  been  as  closely  observed  and  as  thoroughly  determined 
by  experimenters  as  the  limit  of  rupture.  The  latter  was  for- 


TENSION. 


45 


merly  considered  more  important,  and  hence  furnished  the 
basis  for  determining  the  safe  limit  of  the  load.  Observations 
on  good  constructions  have  led  engineers  to  adopt  the  following 
values  as  mean  results  for  permanent  strains  on  bars  :  — 


Further  observations  will  be  made  upon  this  subject  in  the 
latter  part  of  this  volume. 


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THE   RESISTANCE    OF   MATERIALS. 


CHAPTER  II. 

COMPRESSION. 

38.  RESISTANCE  TO  COMPRESSION  also  divides  itself  into 
twro  general  problems — elastic  and  ultimate.  The  law  of  elastic 
resistance  for  compression  may  be  as  readily  found  as  that  for  ten- 
sive resistance  ;  but  the  law  of  resistance  to  crushing  is  very  com- 
plex. If  the  pieces  which  are  subjected  to  this  stress  are  long, 
they  will  bend  under  a  heavy  stress,  unless  they  are  confined,  and 
when  they  bend  they  break  partly  by  bending  and  partly  by  crush- 
ing. If  the  pieces  are  very  short,  compared  with  their  diameter, 
they  may  be  crushed  without  being  bent ;  but  even  in  this  case, 
with  granular  substances,  the  yielding  is  more  or  less  peculiar, 
dividing  off  in  pieces  at  certain  angles  with  the  line  of  pressure. 
The  results  of  some  experiments  will  now  be  given,  which  will 
enable  us  to  test  the  prevailing  theories  upon  this  subject. 

ELASTIC   RESISTANCE. 
TABLE 

Shotting  the,  compression,  permanent  set,  and  coefficient  of  elasticity  *  of  a  solid 
Y+*^~>     cylinder  10  inches  long  and  1.382  inch  diameter. 


Weight  per 
square  inch  of 
section  in  Ibs. 

Compression  per 
inch  of  length. 

Permanent  set  per 
inch  of  length. 

Coefficient  of 
elasticity. 

1,000 

0.000090 

0. 

11,111,000 

2,000 

0.000170 

0. 

11,824,000 

3,000 

0.000255 

0.000005 

11,843,100 

4,000 

0.000320 

0.000015 

12,500,000 

5,000 

0.000385 

0.000025 

12,987,000 

6,000 

0.000455 

0.000030 

13,189,000 

7,000 

0.000505 

0.000035 

13,861,300 

8,000 

0.000575 

0.000045 

13.813,000 

9,000 

0.000645 

0.000055 

13^52,000 

10,000 

0.000705 

0.000070 

14,196,000 

15,000 

0.001035 

0.000170 

14,492,000 

20,000 

0.001395 

0.000300 

14,337,000 

25,000 

0.001825 

0.000495 

13,687,900 

30,000 

0.002380 

0.000820 

12,602,300 

*  The  author  computed  the  coefficients  of  elasticity  from  the  other  data  of 
the  table. 


COMPRESSION.  47 

39.  COMPRESSION  OF  CAST-IRON. — Captain  T.  J.Rodman, 
in  his  report  upon  metals  for  cannon,  page  163,  has  given  the 
results  of  experiments  upon  a  piece  of  cast-iron,  which  was 
taken  from  the  body  of  the  same  gun  as  was  the  specimen  re- 
ferred to  on  page  11  of  this  work,  the  results  of  which  are  given 
on  the  preceding  page. 

We  observe  that  the  coefficient  of  elasticity  is  much  less  for 
the  first  strains  than  for  those  that  follow.  It  thus  appears  that 
this  metal  resists  more  strenuously  after  it  has  been  somewhat 
compressed  than  at  first.  The  coefficient  of  elasticity  is  con- 
siderably less  than  for  the  corresponding  piece,  as  given  on  page 
11.  The  difference  is  very  much  greater  than  that  found  by 
Mr.  Hodgkinson  in  the  specimens  which  he  used  in  his  experi- 
ments. He  took  bars  10  feet  long,  and  about  an  inch  square, 
and  fitted  them  nicely  in  a  groove  so  that  they  could  not  bend, 
and  occasionally,  during  the  experiment,  they  were  slightly 
tapped  to  avoid  adherence.  The  metal  was  the  same  kind  as 
that  used  in  the  experiment  recorded  on  page  13. 


TABLE 

Giving  the  results  of  experiments  by  Mr.  Hodgkinson  on  bars  of  cast-iron  Wfeet 

long. 


Pressure  per 
square  inch  of 
section. 
P. 

Compression  per  inch  of  length. 

Coefficient  of 
elasticity  per 
square  inch. 

Error  in  parts  of 
P  of  the  formula 
P=170,763AC 
—36,318  AO 

Total. 
A0 

Permanent. 

Ibs. 

in. 

in. 

Ibs. 

2064.74 

0.0001561 

0.00000391 

13,231,300 

—    -1-f 

4129.49 

0.0003240 

0.00001882 

12,764,910 

—  IsTlf 

6194.24 

0.0004981 

0.00003331 

12,442,300 

+  -h 

8258.98 

0.0006565 

0.00005371 

12,585,100 

+  T3:n> 

10323.73 

0.00082866 

0.00007053 

12,467,100 

+  TTl2 

12388.48 

0.00100250 

0.00009053 

12,357,200 

+   1F4~8 

14453.22 

0.00128025 

0.00011700 

12,253,700 

+  779 

16517.97 

0.00136150 

0.00014258 

12,141,200 

+  Ts¥ 

18582.71 

0.00154218 

0.00017085 

12,058,100 

+  TJT 

20647.46 

0.00171866 

0.00020685 

12,021,800 

+  lfr§ 

24776.95 

0.00208016 

0.00036810 

11,920,000 

—    ]  7~S 

28906.45 

0.00247491 

0.00045815 

11,687,400 

2  7  o~ 

33030.80 

0.0029450 

0.00050768 

11,222,750 

+  TiV 

37159.65 

0.003429 

THE   RESISTANCE    OF   MATERIALS. 


In  this  case  the  highest  coefficient  of  elasticity  results  from 
the  smallest  strain  which  is  recorded.  The  difference  in  this 
respect  between  this  example  and  the  preceding  one  results 
doubtless  from  the  internal  structure  of  the  iron.  The  coeffi- 
cient in  both  these  cases  is  much  less  than  that  found  for 
other  kinds  of  cast-iron,  as  is  shown  in  the  table  of  resistances 
in  the  Appendix. 

Mr.  Hodgkinson  proposed  the  empirical  formula, — P= 
170,763XC  —  36,318X2,— to  represent  the  results  of  the  experi- 
ments ;  and  although  it  may  represent  more  nearly  the  results 
of  a  greater  range  of  strains  than  equation  (3),  yet  there  is  no 
advantage  in  its  use  in  practice. 

I  O.     COMPRESSION  OF  WROUGHT,  IRON. 

Mr.  Hodgkinson  also  made  experiments  upon  bars  of  wrought 
iron  in  precisely  the  same  manner  as  upon  those  of  cast  iron, 
the  results  of  which  are  given  in  the  following 

TABLE 

Giving  the  results  of  eocperiments  by  Mr.  E.  Hodgkinson  on  bars  of  wrougJit  iron, 
each  of  which  was  ten  feet  long.* 


Weight  producing 
the  compression. 

1st  Bar. 

Sectional.  025  x  1.025  sq.  in. 

2d  Bar. 
Section=1.016xl.02  sq.  in. 

Amount  of 
Compression. 

Value  of  E. 

Amount  of 
Compression. 

Value  of  E. 

Ibs. 

inch. 

Ibs. 

inch. 

Ibs. 

5098 

0.028 

20,796,500 

0.027 

21,864,000 

9578 

0.052 

21,049,000 

0.047 

23,595,000 

14058 

0.073 

21,979,000 

0.067 

24,273,000 

16298 

0.085 

21,343,000 

18538 

0.096 

22,156,000 

0.089 

24,108,000 

20778 

0.107 

22,160,000 

0.100 

24,038,000 

23018 

0.119 

23,587,000 

0.113 

23,587,000 

25258 

0.130 

22,095,000 

0.128 

23,679,000 

27498 

0.142 

22,111,000 

0.143 

22,259,000 

29738 

0.152 

21,938,000 

0.163 

21,139,000 

31978 

0.174 

20,979,000 

0.190 

19,478,000 

In  \  hour. 

0.261 

Again  after  i 

hour. 

0.269 

Then  repeated. 

0.328 

The  coefficients  of  elasticity  were  computed  by  the  author. 


COMPRESSION. 


49 


GRAPHICAL     REPRESENTATION. These    tWO     CaSGS 

are  graphically  represented  in  Fig.  13.  It  is  seen  from  the  tables 
that  the  compressions  are  quite  uniform  for  a  large  range  of 
strains,  and  hence  equation  (2),  page  17,  is  applicable  to  com- 
pressive  strains  when  within  the  elastic  limits.  In  the  case  of 
the  wrousfht-iron  bars,  the  first  one  attains  its  maximum  coeffi- 
cient  of  elasticity  for  a  strain  somewhat  less  than  one-half  its 
ultimate  resistance  to  crushing,  and  the  second  bar  at  about  one- 
third  its  ultimate  resistance. 


30000 
20000 
10000 

/ 

/ 

.ftjL 

2 

7/ 

V. 

,/ 

'/ 

/ 

/ 

// 

// 

// 

/ 

// 

/ 

/ 

/ 

/ 

Us. 

annn 

/ 

/ 

0-02  in.                    0-10                               -20 

FIG.  13. 

43.    COMPARATIVE  RESISTANCE  OF  CAST  AND  WROUGHT 

IRON. — The  coefficient  of  elasticity  is  a  measure  of  the  com- 
pressibility of  metals.  Hence,  an  examination  of  the  two 
preceding  tables  shows  that  of  the  specimens  used  in  these  ex- 
periments, the  cast  iron  was  compressed  nearly  twice  as  much 
as  the  wrought  iron  for  the  same  strains.  An  examination  of 
the  table  of  resistances,  in  the  appendix,  shows  that  for  a  mean 
4 


50  THE   RESISTANCE    OF   MATERIALS. 

value  wrought  iron  is  compressed  about  two-thirds  as  much  as 
cast  iron  for  the  same  strain.  The  same  ratio  evidently  holds 
for  tension.  This  is  contrary  to  the  popular  notion,  that  cast 
iron  is  stiff er  than  wrought  iron ;  for  it  follows  from  the  above 
that  a  cast-iron  bar  may  be  stretched  more,  compressed  more, 
and  bent  more,  than  an  equal  wrought  iron  one  with  the  same 
force  under  the  same  circumstance,  and  in  some  cases,  the 
changes  will  be  twice  as  great.  One  reason  why  cast  is  con- 
sidered stiffer  than  wrought  iron  probably  is,  that  wrought  iron 
does  not  fail  suddenly  as  a  general  thing,  but  it  can  be  seen  to 
bend  for  a  long  time  after  it  begins  to  break ;  while  cast  iron,  on 
account  of  its  granular  structure,  fails  suddenly  after  it  begins, 
and  the  bending  which  has  previously  taken  place  is  not 
noticed.  It  is  not  safe  to  trust  to  such  general  observations  for 
scientific  or  even  practical  purposes,  but  careful  observations 
must  be  made,  so  that  all  the  circumstances  of  the  case  may  be 
definitely  known.  It  will  hereafter  be  shown  that  the  ultimate 
resistance  to  crushing  of  cast  iron  is  double  that  of  wrought 
iron,  and  yet  Fairbairn  and  other  English  engineers  have  justly 
insisted  upon  the  use  of  wrought  iron  for  tubular  and  other 
bridges.  For,  without  considering  the  comparatively  treacherous 
character  of  cast  iron  when  heavily  loaded,  it  appears  that 
within  the  elastic  limits  (and  the  structure  should  not  be  loaded 
to  exceed  that),  a  wrought  iron  structure  is  stiff er  than  a  cast 
iron  one  of  the  same  dimensions,  and  will  sustain  more  for  a 
given  compression,  extension,  or  deflection. 

44.  COMPRESSION  OF  OTHER  MATERIALS. — All  materials 
are  compressible  as  well  as  extensible,  and  it  is  generally 
assumed  that  their  resistance  to  compression,  within  the  elastic 
limits,  is  the  same  as  for  extension ;  but,  as  has  been  seen  in 
the  previous  articles,  this  is  not  rigorously  correct.  Indeed  the 
same  piece  resists  differently  under  different  circumstances, 
depending  upon  its  temperature,  the  duration  of  the  strain,  and 
the  suddenness  with  which  the  force  is  applied.  But  these 
changes  are  not  great,  and  the  mean  value  of  the  coefficient  of 
elasticity  is  sufficiently  exact  for  practical  cases. 


ULTIMATE  STRENGTH. 
MODULUS  FOR  CRUSHING. — The  modulus  of  resistance 


COMPRESSION.  51 

to  crashing  is  the  pressure  which  is  necessary  to  crush  pieces  of 
a  material  whose  length  does  not  exceed  from  one  to  five 
times  its  diameter,  and  whose  section  is  unity.  The  value 
thus  found  we  call  C.  It  is  found  by  experiment  that  the  re- 
sistance of  all  substances  used  in  the  mechanic  arts  varies  very 
nearly  as  the  section  under  pressure.  Hence,  if 

P  =  the  crushing  force,  and 

K  =  the  section  under  pressure,  we  have 

P  =  CK     .....         (22) 
46,  MODULUS  OF  STRAIN  —  If  the  force  P  is  not  sufficient 
to  crush  the  piece,  we  have  for  the  strain  on  a  unit  of  section 


It  is  necessary  to  use  short  pieces  in  determining  the  value  of 
C,  because  long  pieces  will  bend  before  breaking,  and  will  not 
be  simply  crushed,  but  will  break  more  like  a  heam. 


52 


THE   EESISTANCE    OF   MATERIALS. 


47.     RESISTANCE    TO    CRUSHING    OF    CAST-IRON. 

TABLE 

Of  the  results  of  experiments  on  the  tensile  and  crushing  resistances  of  cast  iron  of 
various  kinds,  made  by  Eaton  Hodgkinson* 


Description  of  the  iron. 

Tensile 
Strength  per 
square  inch. 

Height  of 
Specimen. 

Crushing 
strength  per 
square  inch. 

Ratio  of  tenacity 
to  crushing. 

T. 

C. 

T:C. 

Lbs. 

Inch. 

Lbs. 

Mean. 

Low  Moor  Iron,  No.  1 

12,694 

[it 

64,534 
56,445 

1:5-084)  1     .  ~R- 
1  :  4-446  f1:4765 

"         "        No.  2 

15,458 

U 

99,525 
92,332 

1  :  6-438  \  1  .  fi.205 
1:  5-973  f1-6205 

Clyde  Iron,  No.  1  

16,125 

14 

92,869 

88,741 

1  :  5-759  i.,.,0, 
1  :  5-503  f1-6631 

"         "     No.  2  

17,807 

ill 

109,992 
102,030 

1  :  6-177  K.K.OKO 
1:  5-729  f1'5903 

"         "    No.  3.... 

23,468 

14 

107,197 
104,881 

1:4-568).     ...^ 
1  :  4-469  f1'4518 

Blaenavon  Iron,  No.  1 

13,938 

14 

90,860 
80,561 

1  :  6-519  )  1    . 
1  :  5-780  [1:6*149 

«'            u     No.  2 

16,724 

14 

117,605 
102,408 

1  :  7-032  ii.6.577 
1  :  6-123  f1'0577 

"    .       u     No.  3 

14,291 

f.4 

68,559 
68,532 

1:  4-797  }  1.4.706 
1:  4-795  f  1-47Jb 

Calderlron  No.  1  

13,735 

14 

72,193 

75,983 

1:5-256?  1.5.894 
1:  5-532  f1>5dy4 

Coltness  Iron,  No.  3.  . 

15,278 

U 

100.180 
101,831 

1:6-557).    PA11 
l:6-665f1:6611 

Brymbo  Iron,  No.  1  .  . 

14,426 

14 

74,815 
75,678 

1  :  5-186  L,.^, 
1:  5-264  f1'5"510 

u        u     No.  3.. 

15,508 

til 

76,133 
76,958 

1  :  4-909  }  1  .  4.936 
1:  4-963  f1'4 

Bowling  No  2 

13,511 

\     4 

76,132 

J:5'6Si  1-5-555 

I  H 

73,984 

1  :  5  -476  J 

Ystalyf  ea    No  2  

14,511 

5    4 

99,926 

1:6-886)  1>6.735 

(Anthracite) 

1  H 

95,559 

1  :  6-585  \L 

Yniscedwyn,  No.  1  ... 
(Anthracite) 

13,952 

It! 

83,509 

78,659 

1:5-985)  1  .  ..R11 
1:  5-638  f1'5811 

"            No.  2... 

13,348 

j    * 

\  U 

77,124 
75,369 

1:5-778)  ,,^2 
1  :  5-646  f1'5712 

Stirling,  2d  quality.  . 

25,764 

14 

125,333 
119,457 

1:  4-865  i1>4.751 
1:  4-637  f1'4751 

"      3d  quality.. 

23,461 

[4 

158,653 
129,876 

1:  6-762  [  1.6.149 
1  :  5-536  [1<b 

Mean       .   .  . 

16,303 

\  •• 

88,800 

\  •• 

94,730 

Mean  ratio  1  :  5  '64 

*  Supplement  to  Bridges,  by  Geo.  K.  BruneU,  and  Wm.  T.  Clark.   John  Weale, 
London. 


COMPRESSION.  53 

In  this  table  the  ratio  of  resistances  range  from  less  than  4£ 
(Clyde,  No.  3)  to  more  than  7  (Blaenavon,  I£o.£).  The  same  ex- 
perimenter once  obtained  the  ratio  of  8.493  from  a  specimen 
of  Carron  iron,  No.  2,  hot  blast  ;*  and  the  mean  of  several  ex- 
periments, made  at  the  same  time,  gave  6.594.  Hence  we  have, 
as  the  mean  result  of  a  large  number  of  experiments,  that  the 
crushing  resistance  of  cast  iron  is  about  6  times  as  great  as  its 
tenacity ;  but  the  extremes  are  from  4 \  to  8-J-  times  its  tenacity. 

48.    RESISTANCE    OF    WROUGHT    IRON    TO    CRUSHING. — 

Comparatively  few  experiments  have  been  made  to  determine 
how  much  wrought  iron  will  sustain  at  the  point  of  crushing, 
and  those  that  have  been  made  give  as  great  a  range  of  results 
as  those  for  cast  iron.  Wrought  iron  being  fibrous,  does  not  in- 
dicate the  point  of  yielding  as  distinctly  as  cast  iron  and  other 
granulous  substances. 

Hodgkinson  gives  C= 65000  f 

Eondulet          "      C= 70800  J 

Weisbach         "      C= 72000  § 

Eankine  "      C = 30000  to  40000  | 

" — 23  *}££'()"  -i—  t-j  '-==-• 
Hence  it  appears  that  the  crushing  resistance1  of  wrought  i 

is  from  J  to  f  as  much  as  its  tenacity. 

419.  RESISTANCE  OF  WOOD  TO  CRUSHING. — The  resistance 
of  wood  to  crushing  depends  as  much  upon  its  state  of  dry  ness, 
and  conditions  of  growth  and  seasoning,  as  its  tenacity  does. 
The  following  are  a  few  examples : — 


Kind  of  Wood. 

Moderately 
Dry. 

Very  Dry. 

Ash     :  

8,680 

9,360 

Oak  (English)         ....            

6  480 

10058 

Pine  (Pitch)  .           

6  790 

6  790 

These  results,  compared  with  the  corresponding  numbers  in 


*  Resistance  des  Materiaux,  Morin,  p.  95. 

f  Vose,  Handbook  of  Railroad  Construction,  p.  127. 

\  Mahan's  Civil  Engineering,  p.  97. 

§  Weisbach  Mech.  and  Eng.,  vol.  i.,  p.  215. 

|  Rankin's  Applied  Mech.,  p.  633. 


THE   RESISTANCE    OF   MATERIALS. 


article  22,  show  that  these  kinds  of  wood  will  resist  from 
nearly  2  times  as  much  to  tension  as  to  compression. 


to 


RESISTANCE  OF   CAST  STEEL   TO   CRUSHING.  -  Major 

Wade  found  the  following  results  from  experiments  upon  the 
several  samples,  all  of  which  were  cut  from  the  same  bar  and 
treated  as  indicated  in  the  table.* 


Specimen. 

Length. 

Diameter. 

Crushing  in 
Ibs.  per  sq.  inch. 

Not  Hardened  

1-021 

0-400 

198  944 

Hardened,  low  temper  

0-995 

0-402 

354,544 

mean     " 

1-016 

0-403 

391  985 

"         high      "      for  tools  for 
turning  hard  steel.  .  . 

1-005 

0-405 

373,598 

LI  I 

51  .    RESISTANCE  OF  GLASS  TO  CRUSHING.  -  We'owe   most 

of  our  knowledge  of  the  strength  of  glass  to  Wm.  Fairbairn  and 
T.  Tate,  Esq.  According  to  their  experiments  we  have  the 
following  results  for  the  crushing  resistance  of  specimens  of 
glass  whose  height  varied  from  one  to  three  times  their 
diameter. 

MEAN   CRUSHING   EESISTANCE   OF   CUT-GLASS   CUBES   AND   ANNEALED 
GLASS    CYLINDERS. 


Weight  per  Square  Inch. 


Cubes. 

Cylinders. 

Flint  Glass  

Ibs. 

13,130 

Ibs. 

27582 

Green  Glass 

20206 

31  876 

Crown  Glass            

21  867 

31  003 

18,401 

30  153 

The  ratio  of  the  mean  of  the  resistances  is  as  1  to  1*6  nearly. 

The  cylinders  were  cut  from  round  rods  of  glass,  and  hence 
retained  the  outer  skin,  which  is  harder  than  the  interior,  while 
the  cubes  were  cut  from  the  interior  of  large  specimens.  This 


*  Report  on  Metals  for  Cannon,  p.  258. 


COMPRESSION.  55 

may  partially  account  for  the  great  difference  in  the  two  sets  of 
experiments.  The  cubes  gave  way  more  gradually  than  the 
cylinders,  but  both  fractured  some  time  before  they  entirely 
failed.  The  cylinders  failed  very  suddenly  at  last,  and  were 
divided  into  very  small  fragments.  The  specimens  had  rubber 
bearings  at  their  ends,  so  as  to  produce  an  uniform  pressure 
over  the  whole  section. 


STRENGTH  OF  PILLARS. — The  strength  of  pillars  for 
incipient  flexure  has  been  made  the  subject  of  analysis  by  Euler 
and  others,  but  practical  men  do  not  like  to  rely  upon  their 
results.  Mr.  Hodgkinson  deduced  empirical  formulas  from  ex- 
periments which  wrere  made  upon  pillars  of  wood,  wrought  iron, 
and  cast  iron.  The  experiments  were  made  at  the  expense  of 
Wm.  Fairbairn,  and  the  first  report  of  them  was  made  to  the 
Royal  Society,  by  Mr.  Hodgkinson,  in  1840.  The  following  are 
some  of  his  conclusions  : — 

1st.  In  all  long  pillars  of  the  same  dimensions,  wrhen  the 
force  is  applied  in  the  direction  of  the  axis,  the  strength  of  one 
which  has  flat  ends  is  about  three  times  as  great  as  one  with 
rounded  ends. 

2d.  The  strength  of  a  pillar  with  one  end  rounded  and  the 
other  flat,  is  an  arithmetical  mean  between  the  two  given  in  the 
preceding  case  of  the  same  dimensions. 

3d.  The  strength  of  a  pillar  having  both  ends  firmly  fixed,  is 
the  same  as  one  of  half  the  length  with  both  ends  rounded. 

4th.  The  strength  of  a  pillar  is  not  increased  more  than  -^th 
by  enlarging  it  at  the  middle. 

To  determine  general  formulas,  bars  of  the  same  length  and 
different  sections  were  first  used  ;  then  others,  having  constant 
sections  and  different  lengths  ;  and  formulas  were  deduced  from 
the  results.  The  formulas  thus  made  were  compared  with 
the  results  of  experiments  on  bars  whose  dimensions  differed 
from  the  preceding.  The  following  are  the  results  of  some  of  his 

; 


•     •• 


56 


THE   RESISTANCE    OF   MATERIALS. 


EXPERIMENTS    ON    SQUARE   PILLARS. 


Length  of 
the  bars. 

Side  of  the  square. 

Crushing  weight. 

Exponent  of  the 
side. 

Feet. 

Inches. 

Lba. 

10 

0-766 
1-51 

1,948 
23,025  f 

3-57 

10 

1-00 
1-50 

4,225  ) 
23,025  f 

4-17 

n 

1-02 
1-58 

10,236 

45,873  f 

3-69 

7* 

0-50 
1-00 

583) 
9,873  \ 

4-08 

5 

0-50 
1-00 

1,411 

18,038  f 

3-67 

2| 

0-502 
1-00 

4,216 

27,212  ' 

2-69 

3* 

0-502 
0-76 

4,216 
15,946 

3-28 

Mean.  .  .  . 

3-591 

The  fourth  column  is  computed  as  follows  :  — 
Suppose  that  the  strengths  are  as  the  x  power  of  the  diame- 
ters, then  for  the  first  bar  we  have 


* 


0.766 


1948 


or  1.987—  11.30 


=  3.5T. 


log.  1.987 

The  others  are  computed  in  the  same  way. 
An  examination  of  the  table  shows  tljat  when  the 
section  is  the  same  the  strength  varies  inversely,  as. the  length. 
Thus,  of  two  bars  whose  cross  section  is  one  square  inch,  the  one 
five  feet  long  is  nearly  four  times  as  strong  as  the  one  ten  feet 
long. 

Let  I    —  length  of  one, 
I'  =       «      of  other, 
d  =  diameter  of  first  one, 
df  =        "         of  the  second  one,  and 
='  the  power  of  the  length. 


Then  the  strength  of  the  first  one  is,  P=  constant 
"  "  "       second  is,  Pr  =  constant  x 


COMPRESSION. 


57 


in  which  substitute  the  values  from  any  two  experiments.    Thus 

if  we  take  from  the  table 

I'  =  10  feet,  d'  —  \  inch,  P'  =  4225  Ibs.,  and 

I  =    5  feet,  d  =  1  inch,  and  P  =  18038  Ibs.,  we  have 

18038 


4225    ' 
log.  4.2694 


=  2.094. 


Proceed  in  a  similar  way  with  each  of  the  others  and  take 
the  mean  of  the  results  for  the  power  to  be  used.  In  this  way 
was  formed  the  following 

TABLE 

For  the  absolute  strength  of  columns. 

In  which  P  =  crushing1  weights  in  gross  tons, 

d   —  the  external  diameter,  or  side  of  the  column  in  inches, 
di  =  the  internal  diameter  of  the  hollow  in  inches,  and 
I    =  the  length  in  feet. 


Kind  of  Column. 

Both  ends  rounded,  the 
length  of  the  column  ex- 
ceeding fifteen  times  its 
diameter. 

Both  ends  flat,  the  length  of 
the  column  exceeding  thirty 
times  its  diameter. 

Solid  Cylindrical  Columns  of  ) 

TONS. 

/•" 

P1  A.  O 

TONS. 

d3  -5  5 

—  14=.  »     i\  .7 

—  44.10  j1<7 

Hollow  Cylindrical  Columns  ) 

3.76  ,  3.76 

d  —  &i 

/±V8" 

of  cast  iron     ) 

P   -   13             JLT 

_  44.34        Z1  ..- 

Solid  Cylindrical  Columns  of  ) 

£3.76 
P/19 

£1.08 

P1  11  WT 

_  4*3     ^ 

—  lOO.  <0       T2 

Solid  Square  Pillar  of  Dant-  ) 

d* 

PI  n  OK  

Solid  square    Pillar  of    red  ) 

j__  ,qOQi                                               l 

d* 

P  =  7.81  |j 

The  above  formulas  apply  only  in  cases  where  the  length  is 
so  great  that  the  column  breaks  by  bending  and  not  by  simple 
crushing.  If  the  column  be  shorter  than  that  given,  in  the 


58  THE   RESISTANCE   OF  MATERIALS. 

table,  and  more  than  four  or  five  times  its  diameter,  the  strength 
is  found  by  the  following  formula  : 


in  which  P  —  the  value  given  in  the  preceding  table, 

K  =  the  transverse  section  of  the  column  in  square 

inches, 
C  =  the  modulus  for   crushing  in  tons  (gross)  per 

square  inch,  and 

"W  =  the  strength  of  the  column  in  tons  (gross).* 
Experiments  have  been  made  upon  steel  pillars  which  gave 
good  results.f 

53.  WEIGHT  OF  PILLARS.  —  From  the  first  formula  of  the 
preceding  table  we  find 

g*w^g 

a  =  -  -  -- 


The  area  of  the  cross  section  is  J  v  d2,  and  the  volume  m 

inches  =  J£  «•  d*  1.   -  ^Sc  /£  t  t4& 

Cast  iron  weighs  450  pounds  to  the  cubic  foot,  hence  the 

450  450  pr.¥¥  jV.tt 

weight  =  ^-^Q  x  3  x  *d2  x  l=-=-=-ax  3.1416  x  -  1  —  , 

' 


which  by  reduction  gives 

weight  =  0.0152803  (?.  I*  '  5  8  V-W  "  _     (24.) 

If  P  is  given  in  pounds,  this  coefficient  must  be  divided  by 


If  the  pillar  is  hollow  the  section  of  the  iron  is  J  *  (d?  —  ^2), 
.and  if  n  is  the  ratio  of  the  diameters,  so  that  d,=nd  this  be- 
comes 

12 

-  J  «r  d?  (1  —  n*)  ;  and  its  volume  in  inches  =  -7-  *  d?  (1  —  n?)  I; 


*  James  B.  Francis,  C.E.,  has  published  a  set  of  tables  which  gives  the 
'strength of  cast-iron  columns,  of  given  dimensions,  by  means  of  equation  (23), 
-and  also  by  those  in  the  above  table. 
l;-  f  London  Builder,  No.  1211. 


COMPRESSION. 


59 


450 


x  3  x  «r  <?  (1  —  n2)  I. 


and  its  weight  in  pounds  = 

If  the  value  of  d  from  the  second  equation  of  the  first  column 
in  the  preceding  table,  be  substituted  in  the  preceding  equation, 
we  find  the 

weight  in  pounds  = 

25  *  1  -  n* 


32  (2240  x 


(P.  r-' 


(25.) 


Proceeding  in  this  way  with  each  of  the  cases  given  above 
and  we  form  the  following  : 

TABLE 

Of  the  weights  in  pounds  of  pillars  in   terms  of  their  lengtlis  in  feet,  and 
crushing  forces  in  pounds.    JL-  4^- 


Kind  of  Pillar. 


Weight  in  pounds. 


Both  ends  rounded. 
1  >  15  d. 


Both  ends  flat. 
f  >  30  d. 


Solid  Cylindrical  Column 
of  cast  iron. 

Hollow    Cylindrical    Col- 
umns   of    cast    iron, 


if  n=0-98 
if  7i=0-95 
if  7i=0-925 
if  7i=0-90 
if  71=0-875 
if  71=0-85 
if  71=0-80 
71=0-75 

Solid  Cylindrical  Columns 
of  Wrought  Iron, 


Square  Column  of  Dant- 
zic  Oak. 


0:0101645 


.„  ^.,_  l-71a 

^QjQtQSSS^ 


.  s-s 


04M&75582 


(P.I  3-68) 
0-003549688  (R^T. 

^oooooooea 


S3048  (P.Z3-68)1- 


>   £3.M\T 


.V* 


0--005649247  (P.I3-68)^ 


0-00152392 

0-00165855 
/^OV70 

0-00189914  (P.Z3 

0-00211340 
0-00201004 


, 
75)T:tTS 


(Cubic  foot  weighs  47.24 
pounds.  ) 


^  ^ 


60  THE   RESISTANCE    OF   MATERIALS. 

If  the  thickness  of  the  metal  (t)  and  the  external  diameter 
are  given,  n  may  be  found  as  follows:  d— %t= internal  diame- 
ter, hence  n=-?=\  —  *.  For  instance,  if  the  external  diameter 
is  6  inches,  and  the  thickness  f  of  an  inch,  the  internal  diame- 
ter is  5J  inches  and  »=-^=0.875. 

The  iron  used  in  the  preceding  experiments  was  Low  Moor 
No.  2,  whose  strength  in  columns  is  about  the  mean  of  a  great 
variety  of  English  cast  iron,  the  range  being  about  15  per  cent, 
above  and  below  the  values  given  above. 

54.  CONDITION  OF  THE  CASTING. — Slight  inequalities  in 
the  thickness  of  the  castings  for  pillars  does  not  materially  af- 
fect the  strength,  for,  as  was  found  by  Mr.  Hodgkinson,  thin 
castings  are  much  harder  than  thicker  ones,  and  resist  a  greater 
crushing  force.    In  one  experiment  the  shell  of  a  hollow  column 
resisted  about  60  per  cent,  more  per  square  inch  than  a  solid 
one.'*  .  But  the  excess  or  deficiency  of  thickness  should  not  in 
any  case  exceed  25  per  cent,  of  the  average  thickness. f     Thus, 
if  the  average  thickness  is  one  inch,  the  thickest  part  should  not 
exceed  1  \  inch,  and  the  thinnest  part  should  not  be  less  than  f 
of  an  inch. 

It  is  also  found  that  in  large  castings  the  crushing  strength 
of  the  part  near  the  surface  does  not  much  exceed  that  of  the 
internal  parts. 

55.  EXPERIMENTS  MADE  BY  THE  NEW   YORK  CENTRAL 

RAILROAD  COMPANY. — The  immediate  object  of  these  experi- 
ments was  to  determine  the  relative  values  of  different  sorts  and 
forms  of  wrought  iron  of  lengths  greatly  exceeding  their  dia- 
meters, when  subjected  to  longitudinal  compression.  The  pieces 
were  not  in  all  cases  broken,  nor  even  materially  altered  in 
form  by  the  compressions  to  wrhich  they  were  subjected,  the 
experiments  being  generally  discontinued  as  soon  as  the  pro- 
gressive rate  of  flexure  due  to  a  regularly  increased  load  was 
ascertained. 

The  testing  machine  used  in  the  experiments  was  designed  by 
C.  Hilton,  and  was  made  at  the  Company's  carpenter  shop,  at  Al- 
bany, by  order  of  the  Chief  Engineer ;  its  arrangement  and  all  its 
principal  details  were  afterwards  found  to  be  exactly  similar  to 

*  Pfril  Trans.,  1857,  p.  890.  f  Stoney  on  Strains,  vol.  ii.,  p.  206. 


COMPRESSION. 


61 


those  of  the  machine  used  for  the  same  purpose  by  Mr.  Eaton 
Hodgkinson,  at  Manchester,  England  (for  a  description  and  draw- 
ing of  which  see  "  Tredgold  on  the  Strength  of  Wrought  and  Cast 
Iron,"  Weale,  London,  1847),  with  this  difference,  that  the  ma- 
chine made  at  Albany  was  of  wood,  while  that  used  by  Mr. 
Hodgkinson  was  of  iron. 


EXPERIMENT  No.  1. 

Made  upon  a  bar  of  English  Crown  Iron  from  the  works  of  Hawkes,  Crawshay 
&  Co.,  Gateshead,  planed  at  both  ends  and  perfectly  straight,  exactly  8  feet  in 
length,  and  of  cross  section  as  sketched  in  Fig.  14. 


14. 


Weight  applied  in  Ibs. 

Deflection  in  parts  of  an  inch. 

Remarks. 

2,568 

.0 

5,468 

.0 

8,468 

.033 

In  direction  C  D. 

10,448 

.0625 

i 

11,738 

.0833 

i 

12,778 

.15625 

< 

13,778 

.1875 

i 

15,208 

.1875 

i 

16,758 

.25 

i 

20,928 

.3125 

On  the  removal  of  the  above  recorded  load,  the  bar  immediately 
resumed  its  original  form,  having  taken  no  appreciable  set. 
Being  once  more  placed  in  the  machine  the  results  were  : — 


Lbs. 

Deflections  in  A  B. 

Deflections  in  C  D. 

20,248 
25,968 
30,348 

.100 

.300 
Bar  bent  do 

.400 
.600 
able  in  C  D. 

THE   RESISTANCE    OF    MATERIALS. 


EXPERIMENT  !No.  2.  : 


Made  upon  a  bar  of  English  Crown  Iron  from  the  same  works,  planed  at  both 
ends  and  perfectly  straight,  exactly  8  feet  in  length,  and  of  the  cross  section 
shown  in  Fig.  15. 


FIG.  15. 


Weight  applied  in  Ibs. 

Deflection  in  parts  of  an  inch. 

Remarks. 

2,568 

.0 

5,468 

.0 

8,178 

.05 

In  direction  C  D. 

10,918 

.075 

13,668 

.1 

16,468 

.125 

19,218 

.156 

21,968 

.1875 

24,678 

.218 

27,478 

.25 

30,248 

.3125 

No  more  than  30,248  Ibs.  was  placed  upon  this  bar,  the  bar 
being  required  for  use,  and  the  quality  of  the  iron  being  very 
soft  and  easily  bent. 

No  appreciable  set  was  found  upon  the  removal  of  the  above 
load.  On  being  placed  a  second  time  in  the  machine,  and  sub- 
jected to  a  load  of  19,218  Ibs.,  the  flexure  was  observed  to  be 
.125  inch,  this  weight  being  left  on  for  42  hours ;  on  removal 
a  permanent  set  was  observed  of  .01  inch. 


COMPRESSION. 


63 


EXPERIMENT  "No.  3. 

Made  upon  a  bar  of  English  Crown  Iron  from  the  same  works,  8  feet  long, 
planed  at  the  ends,  and  cross  section  sketched  below  in  Fig  16. 


16. 


Weights  applied  in  Ibs. 

Deflection  in  parts  of  an  inch. 

Eemarks. 

2,568 

.0 

Flexure  not  appreciable. 

5,468 

.144 

In  direction  C  D. 

8,178 

.168 

*                a 

10,888 

.192 

u 

13,638 

.242 

<t 

'-16,438 

.288 

a 

19,188 

.314 

a 

21,938 

.360 

a 

24,648 

.408 

u 

27,448 

.480 

If 

30,218 

.612 

it 

The  bar  was  found,  on  removing  the  weights,  to  be  perfectly 
straight.  When  replaced  in  the  machine  the  results  were  as 
follows : — 


Weights. 

Inches  in  C  D. 

Inches  in  A  B. 

16,510 

.5625 

.1875 

24,338 

.625 

.21875 

27,638 

.6875 

.25 

f  30,418 

.75 

.3125 

33,188 

.875 

.3750 

35,948 

1.000 

.3750 

With  35,948  Ibs.  the  bar  bent  double  after  four  minutes. 


04:  THE   RESISTANCE   OF    MATERIALS. 

EXPERIMENT  No.  4. 

Made  upon  a  piece  of  English  Crown  Iron  from  the  same  works,  of  the  section 
slwwn  in  Fig.  17,  planed  at  both  ends  and  exactly  5  feet  in  length. 


B 


Fig.  17. 


Weight  in  Ibs. 

Deflection  in  A  B,  in  parts  of 
an  inch. 

Deflection  in  C  D,  in  parts  of 
ar.  inch. 

2,568 

.00 

.0 

3,038 

.015 

.0 

4,038 

.025 

.020 

5,468 

.060 

.050 

8,228 

.083 

.083 

10,968 

.1 

.1 

13,678 

.1 

.125 

16,448 

.1 

.142 

19,248 

.1 

.150 

21,998 

.083 

.166 

24,708 

.083 

.166 

On  the  application  of  27,508  Ibs.  the  bar  assumed  a  new 
form,  as  shown  in  the  figure ;  the  deflection  in  the  new  direction 
is  designated  as  taking  place  in  x. 


Weight  in  Ibs. 

Deflection  in  A  B,  in 
pai-ts  of  an  inch. 

Deflection  in  C  D,  in 
parts  of  an  inch. 

Deflection  in  a;,  in 
parts  of  an  inch. 

27,508 
30,288 
33,058 
35,818 
39,228 

0.083 
.083 
.083 
.083 
Bar  bro 

.166 
.166 
.166 
.225 
ke  at  y. 

.0875 
.1 
.125 

.1875 

COMPRESSION. 


65 


EXPERIMENT  No.  5. 

A  Bar  of  Angle  Iron,  5  feet  in  length,  planed  at  both  ends  and  quite  straight,  of 
the  cross  section  shouon  in  Fig.  18,  furnished  by  the  Albany  Iron  Works, 
Troy,  of  ordinary  quality. 

\ a 

)C 


FIG.  18. 


Weight  in  Ibs. 

Deflection  A  B. 

Deflection  C  D. 

2,568 

.04 

.05 

3.038 

.06 

.08 

4,038 

.08 

.15 

5,468 

.375 

.25 

8,228 

.500 

.375 

10,968 

.5625 

.375 

13,678 

.5625 

.375 

16,448 

.625. 

.375 

19,248 

Broke  or  bent  double. 

The  general  deflection  in  the  direction  of  the  arrow  could  not 
be  observed  until  8,228  Ibs.  were  applied,  when  it  was  succes- 
sively — 

.156,  .25,  .33,  .5. 


No.  6. 

Made  upon  the  bar  used  in  Experiment  No.  2.  It  being  pre- 
sumed that  the  previous  experiment  had  somewhat  weakened 
this  bar,  it  was  determined  to  break  it.  The  following  weights 
were  used:  — 


Weight  in  Ibs. 


22,048 

30,398 
35,928 


Deflection  in  C  D. 


.126 

.5625 

Bar  bent  nearly  double. 


THE   RESISTANCE   OF   MATERIALS. 


56.  COMPRESSION  OF  TUBES.— BUCKLING. — Wrought  iron 
tubes  when  subjected  to  longitudinal  compress! ve  stresses  may 
yield  by  crushing  like  a  block,  or  by  bending  like  a  beam,  or  by 
buckling.  The  first  takes  place  when  the  tube  is  very  short ;  the 
second,  when  it  is  long  compared  with  the  diameter  of  the  tube  ; 
and  the  last,  for  some  length  which  it  is  difficult  to  assign,  inter- 
mediate between  the  others. 

The  appearance  of  a  tube  after  it  has  yielded  to  buckling  is 
shown  in  Figs.  19  and  20. 

The  experiments  heretofore  made  do 
not  indicate  a  specific  law  of  resistance 
to  buckling  ;  but  the  following  general 
facts  appear  to  be  established : — * 

1.  The  resistance  to  buckling  is   al- 
ways less  than  that  to  crushing ;  and  is 
nearly  independent  of  the  length. 

2.  Cylindrical   tubes   are   strongest ; 
and  next  in  order  are  square  tubes,  and 
then  rectangular  ones. 

3.  Rectangular  tubes,  |        |,  are  not  as 
strong  as  tubes  of  this  form  |    |    |.     The 


FIG.  19. 


FIG.  20. 


tubes  in  bridges  and  ships  are  generally  rectangular  or  square. 


COLLAPSE  OF  TUBES. 

•57 '•  THE  RUPTURE  OF  TUBES  which  are  subjected  to 
great  external  normal  pressure  is  called  "  a  collapse."  The 
Hues  of  a  steam-boiler  are  subjected  to  such  an  external  pres- 
sure, and  in  view  of  the  extensive  use  of  steam  power,  the  subject 
is  very  important.  The  true  laws  of  resistance  to  collapsing  were 
unknown  until  the  subject  was  investigated  by  "Wm.  Fairbairn. 
Experiments  were  carefully  made,  and  the  results  discussed  by 
him  with  that  scientific  ability  for  which  he  is  so  noted.  They 
were  published  in  the  Transactions  of  the  Royal  Society,  1858, 
and  republished  in  his  "  Useful  Information  for  Engineers," 
second  series,  page  1. 


Civ.  Eng.  and  Arch.  Jour.,  vol.  xxviii.,  p.  28. 


COMPEESSION. 


67 


The  tubes  were  closed  at  each  end  and  placed  in  a  strong 
cylindrical  vessel  made  for  the  pur- 
pose, into  which  water  was  ^forced 
by  a  hydraulic  press,  thus  enabling 
him  to  cause  any  desirable  pressure 
upon  the  outside  of  the  tube.  In 
order  to  place  the  tube  as  nearly 
as  possible  in  the  condition  of  a 
flue  in  a  steam-engine,  a  pipe  which 
communicated  with  the  external  air 
was  inserted  into  one  end  of  the  tube. 
This  pipe  permitted  the  air  to  escape 
from  the  tube  during  collapse. 

The  vessel,  pipe,  tube,  and  their 
connections  were  made  practically 
water-tight,  and  the  pressure  indicated 
by  gauges. 

Fig.  21  shows  the  appearance  and 
cross-section  at  the  middle  of  the 
short  tubes  after  the  collapse ;  and 
Fig.  22  of  a  long  one.  Although  no 
two  tubes  appeared  exactly  alike  after 
the  collapse,  yet  the  examples  which 
I  have  selected  are  good  types  of  the 
appearances  of  thirty  tubes  used  in 
the  experiments. 

The  tubes  in  all  cases  collapsed 
suddenly,  causing  a  loud  report.  In  the  first  and  second 
tubes  the  ends  were  supported  by  a  rigid  rod,  so  as  to  pre- 
vent their  approaching  each  other  when  the  sides  were  com- 
pressed. 
The  following  tables  give  the  results  of  the  experiments  : — 


FIG.  22. 


FIG.  81. 


v 


68 


THE    RESISTANCE   OF   MATERIALS. 


TABLE   I. 


Mark. 

No. 

Thickness 
of  Plate, 
inches. 
t. 

Diameter 
in  inches. 
d. 

Length  in 
inches. 
L. 

Pressure  of 
Collapse,  Ibs. 
pr.  sq.  in.  of 
Surface. 
P. 

Product  of 
Pressure 
and  Length. 
P.  L. 

Product  of 
the  Pressure, 
Length,  and 
Diameter. 
P.L.  a.  -p. 

A 

B 
C 
D 
E 
F 

1 

2 

3 
4 
5 
6 

0043 

« 

4 

19 
19 
40 
38 
60 
60 

170 
137 
65 
65 
43 
140* 

3230 
2603 
2600 
2470 
2580 
2800 

10412 
10400 
9880 
10320 

Mean 

2714 

10253 

G 
H 
J 
K 
L 
M 

7 
8 
9 
10 
11 
12 

6 

30 
29 
59 
30 
30 
30 

48f 
47f 
32 
52 
65 
85$ 

1440 
1263 
1888 
1560 
1950 

9 

11328 
9360 
11700 

Mean 

1620 

10796 

N 
0 
P 

13 
14 
15 

«< 
« 

u 

8 

u 
u 

30 
39 
40 

39 
32 
31 

1170 
1248 
1240 

9360 
9984 
9920 

Mean 

1219 

9754 

Q 
R 

16 
17 

« 

a 

10 

u 

50 
30 

19 
33 

950 
990 

9500 
9900 

Mean 

970 

9700 

S 
T 
V 

18 
19 
20 

a 
u 
it 

12-2 
12 

u 

58i 
60 
30 

11-0 
12-5 
22 

643-7 
750 
662 

7850 
9000 
7920 

Mean 

6852 

8256 

*  This  tube  had  two  solid  rings  soldered  to  it,  20  inches  apart,  thus  practically  reducing  it  to 
three  tubes,  as  shown  in  Fig.  23. 


FIG.  23. 

t  The  ends  of  both  were  fractured,  causing  collapse,  perhaps  before  the  outer  shell  had  at- 
tained its  maximum, 
t  A  tin  ring  had  been  left  in  by  mistake,  thus  causing  increased  resistance  to  collapsing. 


COMPRESSION.  69 

58.  DISCUSSION  OF  RESULTS. — By  comparing  the  tubes  of 
the  same  diameter  and  thickness,  but  of  different  lengths,  we 
see  that  the  long  tubes  resist  less  than  the  short  ones ;  hence, 
the  strength  is  an  inverse  function  of  the  length,  and  an  ex- 
amination of  the  seventh  column  shows  that  it  is  nearly  a  sim- 
ple inverse  function  of  the  length.     The  first  of  the  4-inch 
tubes  is  so  much  stronger  than  the  others,  it  may  be  neglected 
in  determining  the  law  of  resistance,  although  it  differs  from  a 
mean  of  all  the  others  by  less  than  -J-  of  the  mean.     An  exami- 
nation of  the  several  cases  indicates  that  we  may  safely  assume 
that  the  resistance  to  collapsing  varies  inversely  as  the  lengths 
of  the  tubes.* 

The  mean  of  the  results  for  the  several  diameters  in  the 
last  column  shows  that  the  resistance  diminishes  somewhat 
more  rapidly  than  the  diameter  increases ;  but  this  includes  the 
error,  if  any,  of  the  preceding  hypothesis.  As  the  power  of 
the  diameter  is  but  little  more  than  unity,  it  seems  safer  to  con- 
clude, for  all  tubes  less  than  12  inches  in  diameter,  as  Fair- 
bairn  does,  that  the  resistance  of  tubes  to  collapsing  varies  in- 
versely as  their  diameters.  \ 

59.  L.AW  OF  THICKNESS. — Experiments  were  also  made 
to  determine  the  law  of  resistance  in  respect  to  the  thickness. 
Comparatively  few  experiments  were  made  of  this  character, 
but  these  few  gave  remarkably  uniform  results.     One  of  the 


*  A  more  exact  law  may  be  found  as  follows: — Let  P  =  the  compressing 
force  per  square  inch ;  C  =  a  constant  for  any  particular  diameter  and 
thickness,  I  =  the  length,  and  n  the  unknown  power. 

O 

P  =  —  for  one  case. 
Z" 
fi 

PI  =  _-    for  another. 


By  means  of  this  equation,  and  any  two  experiments  in  which  the  thickness 
and  diameter  are  the  same,  n  may  be  found,  and  by  using  several  experiments 
a  series  of  values  may  be  found  from  which  the  most  probable  result  can  be 
obtained.  But  in  this  case  the  mean  result  is  so  near  unity,  there  is  no  prac- 
tical advantage  secured  by  finding  it. 


70 


THE   RESISTANCE    OF   MATERIALS. 


tubes  (No.  24),  was  made  with  a  butt  joint,  as  shown  in  Fig. 
24,  and  the  others  with  lap^  joints,  as  in  Fig.  25. 


-Fio.  24.  FIG.  25. 

The  following  are  the  results  of  the  experiments 


TABLE   II. 


Mark. 

No. 

Thickness. 
t. 

Diameter. 
d. 

Length  in 
inches. 
L. 

Pressure  per 
square  inch. 
P. 

P.  L. 

Product. 
L.  P.  a. 
=  P* 

W 
X 
Y 
Z 
JJ 

21 
22 
23 
24 
33 

0-25 
0-25 
014 
014 
0125 

9 
18| 

9 
9 

14* 

37 
61 
37 
37 
60 

(450) 

420 
263 
378 
125 

Uncol 
25620 
9694 
13986 
7500 

apsed. 
480375 
89046 
125874 
108750 

Tubes  Nos.  23  and  24  were  exactly  alike  in  every  respect 
except  their  joints  ;  and  it  appears  that  the  butt  joint,  No.  24, 
is  1*41  times  as  strong  as  the  lap  joint,  a  gain  of  41  per  cent. 
But  this  is  a  larger  gain  than  is  indicated  in  other  cases ;  for 
instance,  No.  33,  which  is  also  a  lap  joint,  offers  a  greater  re- 
sistance as  indicated  in  the  last  column,  than  No.  23,  although 
the  former  is  not  as  thick  as  the  latter.  Still  it  seems  evident 
that  butt  joints  are  stronger  than  lap  joints,  for  with  the 
former  the  tubes  can  be  made  circular,  and  there  is  no  cross 
strain  on  the  rivets,  conditions  which  are  not  realized  in  the 
latter. 

The  resistance  of  the  23d  is  so  small  compared  with  others, 
it  is  rejected  in  the  analysis. 

We  observe  that  the  resistance  varies  as  some  power  of  the 
thickness;  if  then  6yand  n  be  two  constants  to  be  determined 
by  experiment,  and  we  use  the  notation  given  above,  we  shall 
have  for  the  pressure  of  collapse  of  one  tube. 


••tfc*~  A^5s^  /V/^^t^  •< 

J  ^^  COMPRESSION. 


H^c^lfe  ^^t!^ 


and  for  another  tube 

Ct  n 
Pl  =  ^-    .-.4P,L,  =  JP,=  C'C  -    (25) 

Hence  we  Have 

* 


£.  =  (\ 


OT>  n=      =.  ......  (26) 

log.  t  —  log.  ^ 
and  tf=|  =  fk  ......    -    -    -     (27) 

TO    FIND    THE    CONSTANTS    n  AND    £7. 

The  mean  of  the  mean  of  the  values  of  p  from  Table  I.  is 
p  =  %  [10253  +  10796  +  9754+9700  +  8256]:=9752and^O-043. 

Using    these  values  and    others  taken  from  the  preceding 
tables,  and  the  following  values  may  be  found  for  n  :  — 

In  equation  (26)  make^?  =  480375,  t  —  0'25,  p,=  9752,  t,= 
0-043  ;  and  wre  get 

log.  480375  -log.  9752  _ 

log.  o-sS^nbgTo^T 

Similarly,  taking  p  =  480878,*=  O25,^,=  10253,  t,=  0-043  ; 
and  \ve  get 

log.  480375  -  log.  10253  _  0  .  aK 

log.  0-25  -  log.  0-04T" 
The  mean  value  of  p  for  all  but  the  12-inch  tubes  in  Table  I.  is 

p  =  \  (10253  +  10796  -f  9754  +  9700)  =  10125  ; 
hence,  using  p  =  125874,  t  =  0^4,  p,=  10125,  t,=  0-043  ;  and 
we  get 

_  log^5874-log.  10125  _  . 

log.  0-14  -  log.  0-043 

and  taking  p  =108750,  t  =  0-125,  ^=10125  and  t,=  0-043; 
we  get 

log.  108750  -log.  10125 
To~g.  0-125  -  log.  0-043  Jd> 

and  the  mean  of  these  results  is,  n  =  2*18. 

Fairbairn  made  it  2-19  by  including  some  data  which  I  have 
rejected  as  paradoxical  ;  I  have  also  given  more  weight  to  those 


0 


72  THE   RESISTANCE   OF   MATERIALS. 

cases  which  gave  nearly  uniform  results.     The  difference,  how- 
ever, of  0-01  is  too  small  to  seriously  affect  practical  results. 

To  determine  the  constant,  (7,  substitute  the  proper  values 
taken  from  the  preceding  tables  in  equation  (27),  and  we  have 
for  four  cases  the  following  :  — 

9752 

^  ~~    9>298?900- 


=9,864,300. 


The  mean  of  which  is  C=  9,604,150.  Calling  0  =  9,600,000 
and  equation  (2r&  becomes  :  &  /  *-» 

€&  (f  ff       **""      ^^    r        f*    \9  f    ' 

P=  9,600,000  ^j!  -     -     -      (28) 

If  L  be  given  in  feet,  so  that  L  =  12L/?  we  have 

P  =  800,000^4-     -    -    -     ......      (29) 

a.  % 

The  coefficient,  9,600,000,  applies  only  to  the  kind  of  iron 
used  ;  but  the  exponent,  2-18,  is  supposed  to  be  constant  for 
all  kinds  of  iron. 

6O.    FORMULA  FOR  THICKNESS  TO  RESIST  COLLAPSING. 

—  Equation  (28)  readily  gives  the  following  expression  yj? 
vC  ing  the  thickness  in  inches  of  a  tube  to  resist  collapsing  :  — 

J°B£*BJ£UL_.  1-203     -     -     (30) 

' 


61.  ELLIPTICAL  TUBES.  —  Experiments  made  upon  ellipti- 
cal tubes  showed  that  the  preceding  formula  would  give  the 
strength,  if  the  diameter  of  the  circle  of  curvature  at  the  extremity 
of  the  minor  axis  is  substituted  for  d.  The  diameter  of  curva- 

2&a 

ture  is  —  ,  in  which  a  is  the  major  and  5  the  minor  axis. 
o 

Experiments  made  upon  tubes  in  which  the  ends  were  not 
connected  by  internal  rods,  showed  that  the  resistance  was  in- 
versely as  their  length. 

i; 


ce-M^C  4 


,  ce-M^          4 


COMPRESSION.  73 

VERY  LONG  TUBES.  —  Some  experiments  were  made 
upon  a  tube  35  feet  long  and  one  25  feet  long.  Sufficient  pres- 
sure was  applied  to  distort  them,  but  not  to  collapse  them,  and 
it  was  found  that  equation  (28)  erred  by  at  least  20  per  cent., 
giving  too  small  an  amount.  It  was,  however,  very  evident 
that  the  length  was  still  a  very  important  element  in  the 
strength. 

63.    COMPARISON   OF   STRENGTH   FROM   EXTERNAL    AND 

INTERNAL  PRESSURE.  —  Let  p  be  the  internal  pressure  per 
square  inch  at  which  the  tube  is  ruptured,  then  for  tubes  of  the 
same  thickness  and  diameter  we  have  from  equations  (18)  and 
(29),  by  calling  T  =  30,000  Ibs.,  :  ^ 

P~  ~ 


=  P,  then  L  5=  j  ^;J  &•*  8. 

If  t  —  0-25,  then  we  find  L  =  3-56  feet,  that  is,  a  tube  whose 
thickness  is  J  of  an  inch,  and  whose  length  is  3'56  feet,  is  equal- 
ly strong  whether  subjected  to  internal  or  external  pressure. 

If  the  tube  is  so  thick  that  the  unequal  stretching  of  the 
fibres  must  be  considered,  then  equation  (20)  must  be  compared 
with  equal  ion  (29),  in  which  case  we  have  *  —  9.7*  - 


P~  800,000       (r+Jt      c* 
If  p  =  P,  T  =  40,000  Ibs.,  and  2r  =  d  =  4:  inches  ;      £^ 

-+^'18-'f^ 
If  t  =  \  inch,  L  =    5-504  feet. 

If  t  =  l      "    L  =  15       feet. 

64.    RESISTANCE    OF    GLASS    GLOBES    TO     COLLAPSING.  - 

Fairbairn  also  determined  that  glass  globes  and  cylinders  fol- 
lowed the  same  general  law  of  resistance.  For  globes  of  flint 
glass  he  found  : 

P,=28,300,OOOJ^(:(X.  (32) 

and  for  cylinders  of  flint  glass  : 

^=740,000^-     ........      (33) 

Cu  Li 

provided  that  their  length  is  not  less  than  twice,  nor  more  than 


• 

r^^ f^^        ^   L&W^*-  v  IT  V     '-t^  ~^ 

~T~      •  r  ~*t--/          . 

/A^-< 


74  THE   RESISTANCE    OF   MATERIALS. 

six  times  their  diameter.     Dividing  equation  (33)  by  (28)  gives 
Pt      0-0770. 

p~  —  £0-78 

p 

If  t  —  0-043  in.,  p-1  =  0-896 ;  or  the  glass  cylinder  is  nearly  ^ 

as  strong  as  the  iron  one.     If  they  are  equally  strong,  P  =  Pt 
/.  t  =  0-0373  of  an  inch. 


FLEXURE   AND   EtTPTLTEE   FROM   TKANSVEESE    STRESS. 


75 


CHAPTER  III. 


THEORIES  OP  FLEXURE  AND  RUPTURE  FROM  TRANSVERSE 

STRESS. 

65.  REMARK. — The  ancients  seem  to  have  been  entirely 
ignorant   of  the  laws  which  govern  the  strength  of  beams. 
They  made  some  rude  experiments  to  determine  the  absolute 
strength  of  some  solids,  especially  of  stone.     They  may  have 
recognized   some   general  facts  in  regard  to  the  strength  of 
beams,  such  as  that  a  beam  is  stronger  with  its  broad  side  ver- 
tical than  with  its  narrow  side  vertical,  but  we  find  no  trace  of 
any  law  which  was  recognized  by  them.     This  department  of 
science  belongs  wholly  to  modern  times.     A  very  brief  sketch 
of  the  history  of  its  development  is  given  below.* 

66.  GALILEO'S  THEORY. — Galileo  was  the  first  writer,  of 
whom  we  have  any  knowledge,  who  endeavored  to  establish  the 
mathematical  laws  which  govern  the  strength  of  beams,  f     He 
assumed — 

1st.  That  none  of  the  fibres  were  elongated  or  compressed. 

2d.  When  a  beam  is  fixed  at 
one  end,  and  loaded  at  the  other,  it 
breaks  by  turning  about  its  lower 
edge,  B,  Fig.  26 ;  or  if  it  be  sup- 
ported at  its  ends  and  loaded  at  the 
middle  of  the  length,  it  would 
turn  about  the  upper  edge ;  hence 
every  fibre  resists  tension. 

3.  Every  fibre  acts  with  equal 
energy.  From  these  he  readily 
deduced, — that,  when  one  end  is  firmly  fixed  in  a  wall  or  other 


FIG.  26. 


*  For  a  more  complete  history,  see  introduction  to  u  Resistance  des  Corps 
SdKdes,"  par  Navier.     3d  edition.     Paris,  1864. 
f  Opere  di  Galileo.    Bologne,  1856. 


76  THE   RESISTANCE    OF   MATERIALS. 

immovable  mass,  the  total  resistance  of  the  section  is  equal  to 
the  sum  of  all  the  fibres,  or  the  transverse  section,  multiplied 
by  the  resistance  of  a  unit  of  section,  multiplied  by  the  dis- 
tance of  the  centre  of  gravity  from  the  lower  edge.  Hence,  in 
a  rectangular  beam,  if 

T  =  the  tenacity  of  the  material, 

b  =  the  breadth,  and 

d  =  the  depth  of  the  beam  ; 

the  moment  of  resistance  is 


Tbd  x  %d  =  iT&F  -(34) 

67.  ROBERT  HOOKE'S  THEORY.  —  Eobert  Hooke  was  one 
of  the  first,  and  probably  the  first,  to  recognize  the  compressi- 
bility of  solids  when  under  pressure.     In  1678  he  announced 
his  famous  principle,  Ut  tensio  sic  vis  ;  which  he  gave  in  an 
anagram  in  1676,  and  stated  as  the  basis  of  the  theory  of  elasti- 
city that  the  extensions  or  contractions  were  proportional  to  the 
forces  which  produce  them,  and  also  that  when  a  bar  was  bent 
the  material  was  compressed  on  the  concave  side  and  extended 
on  the  convex  side. 

68.  MARRIOTTE'S  AND  LEIBNITZ'S  THEORY.  -  Mamotte, 

in  1680,  investigated  the  subject,  and  finally  stated  the  follow- 
ing principles  :  — 

1st.  The  material  is  extended  on  the  convex  side  and  com- 
pressed on  the  concave  side. 

2d.  In  solid  rectangular  sections  the  line  of  invariable  fibres 
(or  neutral  axis)  is  at  half  the  depth  of  the  section. 

3d.  The  elongations  or  compressions  increase  as  their  distance 
from  the  neutral  axis. 

4th.  The  resistance  is  the  same  whether  the  neutral  axis  is  at 
the  middle  of  the  depth  or  at  any  other  point. 

5th.  The  lever  arm  of  the  resistance  is  f  of  the  depth. 

We  here  find  some  of  the  essential  principles  of  the  resist- 
ance to  flexure,  as  recognized  at  the  present  day  ;  but  the  two 
last  are  erroneous.  As  hereafter  shown,  the  neutral  axis  is  at 
half  the  depth,  and  the  lever  arm  is  f  of  i  the  depth. 

Leibnitz's  theory,  given  in  1684,  was  the  same  as  Marriotte's. 

69.  JAMES  BERNOULLI'S  THEORY  was  essentially  the  same 


FLEXURE   ATCD   RUPTURE   FROM   TRANSVERSE    STRESS.  77 

as  Marriotte's,  except  that  he  stated  that  extensions  and  compres- 
sions were  not  proportional  to  the  stresses.  "  For,"  said  he,  "  if 
it  is  true,  a  bar  might  be  compressed  to  nothing  with  a  finite 
force."  On  this  point  see  Article  16.  He  was  the  first  to  give 
a  correct  expression  for  the  equation  of  the  elastic  curve. 

TO.  PARENT'S  THEORY* — Parent,  a  French  academician 
of  great  merit,  but  of  comparatively  little  renown,  published,  in 
1713,  as  the  result  of  his  labors,  the  following  principles,  in 
addition  to  those  of  his  predecessors  :— 

1st.  The  total  resistance  of  the  compressed  fibres  equals  the 
total  resistance  of  the  extended  fibres. 

2d.  The  origin  of  the  moments  of  resistance  should  be  on 
the  neutral  axis. 

By  the  former  of  these  principles  the  position  of  the  neutral 
axis  may  be  found,  when  the  straining  force  is  normal  to  the 
axis  of  the  beam  ;  and  by  the  latter  he  corrected  the  error  of 
Marriotte  and  Leibnitz ;  showing  that  the  ratio  of  the  absolute  to 
the  relative  strength  is  as  six  times  the  length  to  the  depth,  in- 
stead of  three,  as  will  be  shown  hereafter. 

71.  COULOMB,  IN  1773,  PUBLISHED  the  most  scientific 
work  on  the  subject  of  the  stability  of  structures  which  had 
appeared  up  to  his  time.  He  deduced  his  principles  from  the 
fundamental  equations  of  statics,  and  generalized  the  first  of 
the  principles  of  Parent,  which  is  given  above,  by  saying  that 
the  algebraic  sum  of  all  the  forces  must  he  zero  on  the  three 
rectangular  axes.  This  establishes  the  position  of  the  neutral 
axis  when  the  applied  forces  are  oblique  to  it,  as  well  as  when 
they  are  normal.  He  also  remarked,  that  if  the  proportionality 
of  the  compressions  and  extensions  do  not  remain  to  the  last,  or 
to  the  point  of  rupture,  the  final  neutral  axis  will  not  be  at  the 
centre  of  the  section. 

73.  MODULUS  OF  ELASTICITY.— In  1807  Thomas  Young 
introduced  the  term  modulus  of  elasticity,  which  we  have  defined 
as  the  coefficient  of  elasticity  in  Article  5.  After  this  several 
writers,  among  them  Duhamel,  Navier  in  his  early  writings,  and 
Barlow  in  his  first  work,  stated  the  erroneous  principle,  that  the 
sum  of  the  MOMENTS  of  the  resistances  to  compression  equalled 
those  for  tension. 


78  THE   RESISTANCE   OF   MATERIALS. 

73.  IN  1824  NAVIER  PUBLISHED   the   lectures   which  he 
had  given  to  Vficole  des  Fonts  et  Chaussees,  in  which  he  estab- 
lished more  clearly  those  principles  of  elastic  resistance,  and 
resistance  to  rupture,  which  have  since  his  day  been  accepted 
by  nearly  all  writers.     He  was  the  first  to  show  that  when  the 
stress  is  perpendicular  to  the  axis  of  the  beam,  the  neutral  axis 
passes  through  the  centre  of  gravity  of  the  transverse  sections. 
His  most  important  modifications  in  the  analysis  was  in  making 
ds  =  dx,  or  otherwise,  considering  th&tfor  small  deflections  the 
tangent  of  the  angle  which  the  neutral  axis  makes  with  the 
original  axis  of  the  'beam  is  so  small  compared  with  unity  tJiat 
it  may  be  neglected  ;  and  also,  that  the  lever  arm  of  the  force 
remains  constant  during  flexure.      These  principles  we  have 
used  in  Chapter  V.     He  resolved  many  problems  not  before 
attempted,  and  became  an  eminent  author  in  this  department 
of  science. 

74.  THE  COMMON  THEORY. — The  theories  of  flexure  and 
of  rupture  which  result  from  these  numerous   investigations, 
I  wrill  call,  for  convenience,  the  common   theory.     It  consists 
of  the  following  hypotheses  : — 

1st.  The  fibres  on  the  convex  side  are  extended,  and  on  the 
concave  side  are  compressed,  and  there  are  no  strains  but  com- 
pression and  extension. 

2d.  Between  the  extended  and  compressed  fibres  (or  elements) 
there  is  a  surface  which  is  neither  extended  nor  compressed,  but 
retains  its  original  length,  and  which  is  called  the  neutral  surface, 
or  in  reference  to  a  plane  of  fibres  it  is  called  the  neutral  axis. 

3d.  The  strains  are  proportional  to  their  distance  from  the 
neutral  axis. 

4th.  The  transverse  sections  which  were  normal  to  the  neu- 
tral axis  of  the  beam  before  flexure,  remain  normal  to  the  neu- 
tral axis  during  flexure. 

5th.  A  beam  will  rupture  either  by  compression  or  extension 
when  the  modulus  of  rupture  is  reached. 

6th.  The  modulus  of  rupture  is  the  strain  at  the  instant  of 
rupture  upon  a  unit  of  the  section  which  is  most  remote  from 
the  neutral  axis  on  the  side  which  first  ruptures.  This  is 
called  R 

The  remainder  of  this  article  properly  belongs  to   Chapter 


FLEXURE   AND   RUPTURE   FROM   TRANSVERSE    STRESS.  79 

VI.,  but  it  is  given  here  so  that  the  reasons  for  Barlow's  theory 
may  be  understood. 

If  a  beam  ruptures  on  the  convex  side,  it  appears  that  it 
ought  to  break  when  its  tenacity  (T)  is  reached ;  but  it  is  found  by 
experiment  that  in  this  case  R  always  exceeds  T.  Similarly,  it 
would  seem  that  if  it  failed  by  crushing  on  the  concave  side,  as 
in  the  case  of  rectangular  cast-iron  beams,  R  ought  to  equal  C, 
but  experiment  shows  that  in  this_£ase  J£-exeeeds  C  ;  and  gen- 
erally the  value  of  His  always  between  those  of  T  and  C  for 
the  same  material,*  being  greater  tha^-^re--STrraftgrrgm^-looo  than- 

'  o    O 

&&}mgt*ry  The  values  of  R  in  the  tables  were  deduced  from 
experiments  upon  rectangular  beams,  as  will  hereafter  be 
shown  ;  and  hence,  if  the  common  theory  is  correct,  R  should 
equal  the  value  of  the  lesser  resistance,  whether  it  be  for  com- 
pression or  extension ;  but  it  does  not.  This  discrepancy  be- 
tween theory  and  the  results  of  experiment  *  has  led  Barlow 
to  investigate  the  subject  further,  and  it  has  resulted  in  a 
new  theory  which  he  calls  "  Resistance  to  Flexure  " — an  ex- 
pression which  I  consider  unfortunate,  as  it  does  not  express  his 
idea.  "  Longitudinal  Shearing  "  would  express  his  idea  better, 
as  will  appear  from  the  following  article  : — 

75.  BARL.OWS  THEORY. — According  to  the  common  theory 

*  Mosley's  Mech.  and  Arch. ,  p.  557.  * '  The  elasticity  of  the  material  has  been 
supposed  to  be  perfect  up  to  the  instant  of  rupture,  but  the  extreme  fibres  are 
strained  much  beyond  their  elastic  limits  before  rupture  takes  place,  while  the 
fibres  near  the  neutral  axis  are  but  slightly  strained,  and  hence  the  law  of  pro- 
portionality is  not  maintained,  and  the  position  of  the  neutral  axis  is  changed, 

TJT 

and  the  sum  of  the  moments  is  not  accurately  —  (see  equation  171).     To  de- 

0,1 

termine  the  influence  of  these  modifications  we  must  fall  back  upon  experiment, 
and  it  has  been  found  in  the  case  of  rectangular  beams  that  the  error  will  be 

corrected  if  we  take  —  T  (=  R)  instead  of  T,  where  m  is  a  constant  depend - 
m 

ing  upon  the  material." 

Weisbach,  vol.  ii.,  4th  ed.,  p.  68,  foot-note  says,  "  Excepting  as  exhibiting 
approximately  the  laws  of  the  phenomena,  the  theory  of  the  strength  of  mate- 
rials has  many  practical  defects." 

In  the  Report  of  the  Ordnance  Department,  by  Maj.  Wade,  p.  1,  it  says: — 
"  A  trial  was  made  with  cylindrical  bars  in  place  of  square  ones.  These  gen- 
erally broke  at  a  point  distant  from  that  pressed,  and  the  results  were  so  ano- 
malous that  the  use  of  them  was  soon  abandoned.  The  formula  by  which  the 
strength  of  round  bars  is  computed  appears  to  be  not  quite  correct,  for  the  unit 
of  strength  in  the  round  bars  is  uniformly  much  higher  than  in  the  square  bars 
cast  from  the  same  iron." 


80 


THE   RESISTANCE    OF   MATERIALS. 


the  resistance  at  a  section  is  the  same  as  if  the  fibres  acted  in- 
dependently of  each  other,  and  the  transverse  section  remained 
normal  to  the  neutral  axis.  But  Barlow  correctly  considers 
that  in  order  to  keep  the  transverse  sections  normal  to  the 
neutral  axis,  the  consecutive  planes  of  fibres  must  slide  over 
each  other,  and  to  this  movement  they  offer  a  resistance. 

He  presented  his  view  to  the  Royal  Society  (Eng.),  in  1855, 
and  it  has  since  been  published  in  the  Civil  Engineer  and 
Architect's  Journal,  vol.  xix.,  p.  9,  and  vol.  xxi.,  p.  111.*  The 
subject  is  there  discussed  in  a  very  able  and  thorough  manner, 
and  although  he  may  have  failed  to  establish  his  theory,  by  not 
taking  into  account  all  the  incidents  which  exist  at  the  instant 
of  rupture,  yet  the  results  of  his  analysis  seem  to  agree  more 
nearly  with  the  results  of  experiment  than  those  obtained  by 
any  other  theory  heretofore  proposed. 

It  is  admitted  in  this  theory  that  a  beam  will  rupture  when 
the  stress  upon  any  fibre  equals  its  tenacity,  or  its  resistance  to 
compression,  as  the  case  may  be.  But,  on  the  other  hand,  when 
the  adjacent  fibres  are  unequally  strained,  as  they  are  in  the 
case  of  flexure,  it  requires  a  greater  stress  to  produce  this 
strain  than  it  would  if  the  fibres  acted  independently,  ac- 
cording to  the  previously  assumed  law.  This, 
Barlow  makes  evident  from  the  following 
example : — 

If  a  weight,  P,  Fig.  2V,  is  suspended  on 
a  prismatic  bar,  BCEF,  all  the  fibres  will 
be  equally  strained,  and  hence  equally  elon- 
gated. 

But  if  the  bar  ABCD  be  substituted  for 
the  former,  and  the  weight  P  acts  upon  a  part 
of  the  section,  as  shown  in  the  figure,  it  is  evi- 
dent that  all  the  fibres  will  not  be  equally 
strained,  and  hence  will  not  be  equally  elon- 
gated ;  and  if  the  force  P  was  just  sufficient  to 
rupture  the  bar  FBCE,  it  will  not  be  sufficient 
to  rupture  the  bar  ABCD,  although  P  acts 
directly  upon  the  same  section,  for  the  cohe-  Fia  27 


*  Civ.  Eng.  and  Arch.  Jour.,  Vol.  xix.,  p.  9,  Barlow  says  that  the  strength 
of  a  cast-iron  rectangular  bar,  as  found  from  existing  theory,  cannot  be  recon- 


rqn: 


'. 


FLEXURE   AND   RUPTURE   FROM   TRANSVERSE    STRESS.  81 

sion  of  the  particles  along  FE  will  not  permit  the  fibres  next 
to  that  line  to  be  elongated  as  much  as  if  the  part  AFED 
were  removed ;  and  these  fibres  will  act  upon  those  adjacent,  and 
so  on,  till  they  produce  an  effect  upon  BC.  From  this  we 
see  that  it  takes  a  greater  weight  than  P  acting  upon  the  section 
EC  to  produce  a  strain  T  per  unit  of  section,  when  the  part 
ADEF  is  added.  It  is  also  evident  that  if  the  section  of 
ABCD  is  twice  as  great  as  FBCE,  it  will  not  take  twice  P  to 
rupture  the  fibres  on  the  side  BC. 

A  phenomenon  similar  to  this  takes  place  in  transverse 
strain.  One  side  is  compressed  and  the  other  elongated ;  and 
the  fibres  less  strained  aid  those  which  are  more  strained  by 
virtue  of  the  cohesion  which  exists  between  them,  and  it  takes 
a  greater  force  to  cause  a  strain,  T,  longitudinally  upon  the 
fibres  than  it  would  if  there  were  no  cohesion. 

There  is,  then,  at  the  time  of  the  rupture  of  a  beam,  a  tensile 
strain  on  the  extended  fibres,  and  a  compressive  strain  on 
the  other  fibres,  and  a  longitudinal  shearing  strain  between  the 
fibres,  due  to  cohesion.  These  remarks  will,  I  trust,  enable  the 
reader  to  understand  the  difference  between  the  "Common 
Theory"  and  "  Barlow's." 

Barlow's  Theory  consists  of  the  following  hypotheses : — 

1st.  The  fibres  or  elements  on  the  convex  side  are  extended, 
and  on  the  concave  side  compressed. 

2d.  There  is  a  neutral  surface,  as  in  the  common  theory. 

3d.  The  tensive  and  compressive  strains  on  a  fibre  are  pro- 
portional to  the  distance  of  the  fibre  from  the  neutral  axis. 

4th.  That  in  addition  to  these  there  is  a  "  Resistance  to 
flexure  "  or  longitudinal  shearing  strain,  which  consists  of  the 
following  principles : — 

a.  It  is  a  strain  in  addition  to  the  direct  extensive  and  com- 
pressive forces,  and  is  due  to  the  lateral  cohesion  of  the  adja- 


ciled  with  the  results  of  experiment  if  the  neutral  axis  be  at  the  centre  of  the 
sections.  He  then  proceeded  to  show  by  experiment  that  the  neutral  axis  is 
at  the  centre,  and  then  remarked  that  the  formula  commonly  used  for  a  beam 

2  Tbd2 
supported  at  the  ends  and  loaded  in  the  middle,  or  W  =  .3  — , —  did  not  give 

half  the  actual  strength  if  T  is  the  tenacity  of  the  iron.  He  then  pro- 
ceeds to  point  out  a  new  element  of  strength,  which  he  caljs  "Resistance 
to  Flexure." 


82  THE   RESISTANCE   OF   MATERIALS. 

cent  surfaces  of  fibres  or  particles,  and  to  the  elastic  reaction 
which  ensues  when  they  are  unequally  strained. 

b.  It  is  evenly  distributed  over  the  surface,  and  consequently 
within  the  limits  of  its   operation  its  centre  of   action  will 
be  at  the  centre  of  gravity  of  the  compressed  or  of  the  ex- 
tended section.     This  force  for  solid  beams  Barlow  calls  <p,  and 
for  T  °r  I  sections,  or  open-built  beams,  it  is  easily  deduced 
from  the  following  principle : — 

c.  It  is  proportional  to  and  varies  with  the  inequality  of 
strain  between  the  fibres  nearest  the  neutral  axis   and  those 
most  remote. 

From  this  it  appears  that  if  d'  is  the  depth  of  the  horizontal 
flanges  of  the  I  section,  and  dt  the  distance  of  the  most  remote 

fibre  from  the  neutral  axis,  then  the  resistance  to  flexure  of  the 
jp 

flanges  will  be  <P  -r  and  similarly  for  other  forms. 

5.  Sections  remain  normal  to  the  neutral  axis  during  flex- 
ure. 

6.  Rupture  of  solid  beams  takes  place  when  the  strain  on  a 
unit  of   section  is  T-f-  p,  or  C  4-  p,  whichever  is  smaller,   or 
rather,  whichever  value  is  first  reached. 

V 

76.  REMARKS  UPON  THE  THEORIES. — For  scientific  pur- 
poses it  is  desirable  to  determine  the  correct  theory  of  the 
strength  of  beams,  but  the  phenomena  are  so  complex  that  it 
is  not  probable  that  a  single  general  theory  can  be  found 
which  will  be  applicable  to  all  the  irregular  forms  of  beams 
used  in  practice.  Although  Barlow's  theory  appears  plausible, 
yet  according  to  principle  c  the  resistance  to  flexure,  p,  can- 
not be  uniform  over  the  surface,  as  stated  in  principle  &,  because 
the  proportionality  of  the  elongations  and  compressions  do  not 
continue  up  to  the  point  of  rupture.  The  common  theory  is 
faulty  beyond  what  has  already  been  said  in  the  I  section  ;  for 
in  the  upper  and  lower  portions  the  strains  on  all  the  fibres  are 
not  proportional  to  their  distances  from  the  neutral  axis,  to 
realize  which  the  material  should  be  continuous ;  and  Barlow's 
theory  is  defective  in  the  same  case,  on  account  of  the  peculiar 
strains  upon  the  fibres  at  the  angles  where  the  parts  join.  For 
rupture,  then,  we  can  use  these  theories  to  ascertain  general  facts, 


FLEXURE   AND   RUPTURE   FROM   TRANSVERSE    STRESS.  83 

and  make  the  results  safe  in  practice  by  using  a  proper  coeffi- 
cient of  safety ;  but  for  flexure  the  common  theory  is  suffi- 
ciently exact  if  the  elastic  limit  is  not  passed,  and  this  is  for- 
tunate as  the  conditions  of  stability  should  be  founded  on  the 
elastic  properties  rather  than  on  the  ultimate  strength  of  the 
material.  For  the  rupture  of  rectangular  beams  the  common 
theory  will  be  sufficiently  exact  if  the  value  of  R  is  used  instead 
of  T  or  C  in  the  formulas. 


POSITION  OF  THE  NEUTRAL  AXIS. 
77.    fSITION  FOUND  EXPERIMENTALLY.  -  According    to 

Galileo's,  Marriotte's,  and,  Leibnitz's  theories,  the  neutral  axis  is  on 
the  surface  opposite  the  side  of  rupture. 

Professor  Barlow  made  the  following  experiments  :  —  He  took 
a  cast-iron  beam  and  drilled  holes  in  its  side,  into  which  were  fit- 
ted iron  pins.  He  carefully  measured  the  distance  between  the 
pins,  before  and  after  flexure,  by  means  of  a  micrometer,  and  thus 
found  that  in  solid  cast-iron  beams  bent  by  a  normal  pressure 
the  neutral  axis  passes  through  the  centre  of  the  sections  (Civ. 
Eng.  Jour.,  vol.  xix.,  p.  10).  He  also  made  the  same  kind  of 
an  experiment  on  a  solid  rectangular  wrought-iron  beam,  and 
with  the  same  result  (Civ.  Eng.  Jour.,  vol.  xxi.,  p.  115). 

Some  years  previous  to  the  preceding  experiments,  he  took  a 
bar  of  malleable  iron  and  cut  a  trans  verse  groove  in  one  side,  into 
which  he  nicely  fitted  a  rectangular  key.  When  it  was  bent,  the 
fibres  on  the  concave  side  were  compressed,  and  the  groove  made 
narrower,  so  that  the  key  would  no  longer  pass  through,  and  thus 
he  showed  that  the  neutral  axis  was  between  -J  and  -J-  the  depth 
of  the  beam  from  the  compressed  side  (Barlow's  Strength  of 
Materials,  p.  330  ;  Jour.  Frank  Inst.,  vol.  xvi.,  2d  series,  p.  194). 

Experiments  made  at  the  Conservatoire  des  Arts  et  Metiers, 
in  1856,  on  double  T  sections,  show  that  it  passes  through  the 
centre  of  the  sections  (Morin,  Resistance  des  Materiauv,  p. 
137).  And  experiments  made  at  the  same  time  on  rectangular 
wooden  beams  showed  that  it  passed  at  or  very  near  the  centre 
of  gravity  of  the  sections. 

In  these  experiments  the  elasticity  of  the  material  was  not 
seriously  damaged  by  the  strains.  To  render  them  complete, 

£3t^- 


I&T* 


84  THE   RESISTANCE    OF    MATERIALS. 

the  strains  should  have  been  carried  as  near  to  the  point  of  rup 
ture  as  possible. 

78.    POSITION    DETERMINED  ANALYTICALLY.  -  We  knOW 

from  statics  that  the  algebraic  sum  of  all  the  forces  on  each  of 
the  rectangular  axes  must  be  zero  for  equilibrium  ;  hence,  if  the 
deflecting  forces  are  normal  to  the  axis  of  the  beam,  the  sum  of 
the  resistances  to  compression  must  equal  those  for  tension. 

1st.  Suppose  that  the  coefficient  of  elasticity  for  compression 
equals  that  for  tension.  Then  will  the  compressions  and  exten- 
sions  be  equal  at  equal  distances  from  the  neutral  axis.  In  Fig. 
28,  let  Rc  be  the  strain  on  a  unit  of  fibres  most  remote  from 
the  neutral  axis  on  the  compressed  side,  and  dc  =  the  distance 
of  the  most  remote  fibre  on  the  same  side  ;  then, 

z_£.—  s  =  strain  at  a  unit's  distance  from  the  neutral  axis. 
dc 

Let  k^  ^2)  ^3)    &c-)  be  the  sections  of  fibres  on  one  side  of  the 
neutral  axis,  at  distances  of 

2/D  2/2)  2/3)    &c-)  from  the  axis,  and 

&',  kf  ',  k"'  ',  &G.J  and  y',  y'1  ',  y"'  ',  &c.,  corresponding  quan- 
tities on  the  other  side. 


Then  *(%,  +  %,+%.  +  &<!.)  =s  (k'y'  +  tify"  +V"yf 

or,  %1+&22/2+%s+&c.-  (k'yf  +#y/+>fc/'y/'+&c.)  =  o, 

or,  2%  =  °     ........ 


or  the  neutral  axis  passes  through  the  centre  of  gravity  of  the 
sections.* 

If  the  resistance  to  compression  is  greater  than  for  tension, 
the  neutral  axis  will  be  nearer  the  compressed  side  than  when 
they  are  equal. 

2.  Suppose  that  the  coefficient  of  elasticity  is  not  the  same 
for  tension  as  for  compression. 


*  The  analytical  expression  for  the  ordinate  to  the  centre  of  gravity  is 

fc'ffi  +  foy»  +  &c-  k'v'  +  k"y"  +  &c->       -f 


+  &c.  +  k'  +  k"  -\-  &c.  fydx 

^ 


<t 
XD 


FLEXURE   AND   RUPTURE    FROM   TRANSVERSE    STRESS. 


85 


Let  Fig.  28  represent  the  beam.     Suppose  that  the  sections 

CM  and  EF  were  parallel  be- 
fore deflection.  If  through  IS", 
the  point  where  EF  intersects 
the  neutral  axis,  KH  is  drawn 
parallel  to  CM,  the  ordinates 
between  EF  and  KH  will  re- 
present the  elongations  on  one 
side,  and  the  compressions  on 
the  other,  for  those  fibres 


FIG. 


whose  original  length  was 


Let  1  = 

A   =  Jce  =  the  elongation  of  a  fibre  at  Jc  ; 

p  —  a  pulling  or  pushing  force  which  would  produce  A 

y  =  N£  =  distance  of  any  fibre  from  the  neutral  axis  ; 

k  —  section  of  any  fibre  ; 

Ef  —  coefficient  of  elasticity  for  tension ;  and 

Ec  —  "  "         compression. 

From  equation  (3)  we  have, 


r  — 


I 


But  A  is  directly  proportional  to  its  distance  from  the  neutral 
axis  ;  hence,  if  c  be  a  constant  quantity,  whose  value  rnay  or 
may  not  be  known,  we  shall  have  A  —  cy 


(36) 


Or,  if  we  adopt  the  same  notation  as  in  the  preceding  case,  we 
shall  have  for  the  total  force  tending  to  produce  extension, 

^  =  ^(%,  +  ^  +  %s  +  &c.)    -    -    (37) 
Similarly  for  compression  , 

£p  =  £*S  (*y  +  k"y"  +  K"y'"  +  &c.)         (38) 

Placing  these  equal  to  each  other  and  we  have, 
E<(%i  +  %,  +  %3  +  &c.)  =  Ec  (ky  +  k"y"  +  k'"y 
or,  in  the  language  of  the  integral  calculus, 

E,  *Qydydx  =  Ec  S°_    ydydx,  -  -  (39) 


86 


THE   RESISTANCE    OF   MATERIALS. 


in  which  y  is  an  ordinate  and  x  an  abscissa.  Equation  (39) 
enables  us  to  find  the  position  when  the  form  of  section  is 
known.  In  most  cases,  however,  the  reduction  is  not  easily 
made. 

Example. — Suppose  the  sections  are  rectangular. 

Let  I  =  AC, 
d-  AB, 

y  =  AE  for  the  superior  limit. 
Then  equation  (39)  becomes  pIO 

/•*  r^  r^  /*^~y 

a  I       I    ydydx  =  i        i        ydydx,  which  reduced  becomes 

•70    «/  0  J  Q    J  I)  • 


d 
1+ 


(40) 


a  =  x>,y  =  0 
a  —  0,  y  =  d. 

If  y  is  known  in  equation  (40),  the  ratio  of  the  coefficients  may  easily  be 
found;  for,  we  have  from  (40) 

3d.  Suppose  that  the  deflecting  force  is  not  perpendicular  to 
the  axis,  and  Ec  =  E*  =  E. 

Let  6  —  the  angle  which  P  makes  with  the  axis  of  the  beam 
Fig.  30; 

Px  =  P  cos  6  =  the  com- 
ponent of  P  in  the  direction 
of  the  axis  of  the  beam; 

P2  =  P  sin  6  =  the  com- 
ponent of  P  perpendicular^ 
to  the  axis  of  the  beam ; 

h  =  the  distance   of    the  FIG.  so. 

neuti  al  axis  from  the  centre  of  gravity  of  the  section  AB,  and 
K  —  the  transverse  section. 


FLEXURE   AND   RUPTURE   FROM   TRANSVERSE    STRESS.  87 

The  whole  force  of  compression  equals  the  whole  force  of 
extension,  equations  (37)  and  (38). 


/.P  cos0  +  —^ 

But  the  ordinate  to  the  centre  of  gravity  is  (see  foot-note  on 
page  84), 

,  ~l 


.-.  P  cos  6  =  ~lUi 

•pi 
or  h  =  -         cos  *    .......       (42) 


If  6  =  90°,  h  =  0  as  before  found. 

If  6  =  0  there  is  no  neutral  axis,  for  the  force  coincides  with 
the  axis  of  the  beam.  The  equation  would  show  the  same  re- 

sult, if  the  value  of  c  =  -  =  -,  equation  (45),  were  substituted 

y       I 

in  the  formula,  for  then  p  would  be  infinite,  for  c  —  0,  and 
h  becomes  infinite. 

4th.  Let  the  law  of  resistance  be  according  to  Barlow's 
theory  of  flexure,  and  the  deflecting  forces  normal  to  the  axis 
of  the  beam. 

Using  the  same  notation  as  before,  also 

<#!  —  the  distance  of  the  most  remote  fibre  frcon  the  neutral 
axis,  and 

Q  =  the  coefficient  of  longitudinal  shearing  stress. 

rv 
Then  <p  J  y  dx  =  the  resistance  to  shearing  for  tension, 

o 

r° 
and  £  /     y  dx  =  the  resistance  to  shearing  for  compression, 

47  -y 

and,  proceeding  as  we  did  to  obtain  equation  (39),  we  have 

d~ff  ydyd®  +  ^Sl^^dJj     ydydx  +  vf  y  dx.  (43) 

Examples.  —  Let  the  sections  be  rectangular,  b  —  the  breadth,  d  =  the  depth. 
Then  (43)  becomes 

,  =  -~  (d  -  dtf  +  $  (d-d,) 


Td 


88 


THE   RESISTANCE    OF   MATERIALS. 


the  former  only  of  which  is  admissible. 

If  the  section  is  a  double  T»  as  **  Fi£-  31»  with 
the  notation  as  in  the  figure,  0  will  be  used  in  find- 
ing the  resistance  of  the  vertical  rib,  and  according 

to  Article  75,  0-=—^     of  the    lower    flange,    and 

Cv  —  Cv\ 

0  ^    of   the    upper  flange. 


i, 


FIG.  31. 


It  appears  from  these  several  cases  that  the  neutral  axis 
passes  near  the  centre  of  gravity  in  most  practical  cases,  and  it 
will  be  assumed  that  it  passes  through  the  centre  unless  other- 
wise stated. 


SHEARING   STRESS.  89 


CHAPTEE  1Y. 

[SHEARING  STRESS. 

79.  GENERAL  STATEMENT. — Two  kinds  of  shearing  stress 
are  recognized — longitudinal  arid  transverse — both  of  which  have 
been  defined  in  Article  2.     Materials  under  a  variety  of  cir- 
cumstances are  subjected  to  this  stress — such  as,  rivets  in  shears ; 
the  rivets  in  riveted  plates;  pins  and  bolts  in  spliced  joints; 
beams  subjected  to  transverse  strains;  bars  which  are  twisted; 

/'and,  in  short,  all  pieces  which  are  subjected  to  any  kind  of  distor- 
[  sive  stress  in  which  all  parts  are  not  equally  strained.  In  the 
first  examples  above  enumerated,  all  parts  of  the  section  are 
supposed  to  be  equally  strained.  Shearing  may  take  place  in 
detail,  as  when  plates  or  bars  of  iron  are  cut  with  a  pair  of 
shears,  \vhen  only  a  small  section  is  operated  upon  at  a  time ;  or 
it  may  be  so  done  as  to  bring  into  action  the  whole  section  at  a 
time,  as  in  the  process  of  punching  holes  into  metal,  where  the 
whole  surface  of  the  hole  which  is  made  is  supposed  to  resist 
uniformly. 

80.  MODULUS  OF  SHEARING — The  modulus  of  resistance 
to  shearing  is  the  resistance  which  the  material  offers  per  unit 
of  section  to  being  forced  apart  when  subjected  to  a  shearing 
stress. 

This  we  call  Ss.  The  resistance  for  both  kinds  of  shearing 
has  been  found  to  vary  directly  as  the  section ;  so  that  if 
K  =  the  area  of  the  section  subjected  to  this  stress  the  total  re- 
sistance will  be 

K.&. 

The  value  of  Ss  has  been  found  for  several  substances,  the 
principal  of  which  are  as  follows : — 


90  THE  RESISTANCE  OF  MATERIALS. 

METALS. 

Ss  in  Ibs.  per  square 
inch. 

Fine~"cast  steel  *  92,400 

Eivet  steel  f  64,000 

Wrought  iron  *  50,000 

Wrought-iron  plates  punched  $  51,000  to  61,000 
Wrought  iron  hammered  scrap  punched  §          44,000  to  52,000 

Cast  iron  30,000  to  40,000 

Copper  ||  33,000 

WOOD. 

With  the  fibres. 

White  pine     -  480 

Spruce  470 

Firf      -  592 

Hemlock**     .    -  540 

Oak  780 

Locust  1,200 

Across  the  fibres. 

Eed  pine      -  500  to  800 

Spruce  600 

Larch  ft  970  to  1,700 

Treenails,  English  oak  ft  3,000  to  5,000 

It  will  be  seen  from  these  results  that  the  shearing  strength 
of  wrought  iron  is  about  the  same  as  its  tenacity  ;  of  cast  steel 
it  is  a  little  less  than  its  tenacity ;  of  cast  iron  it  is  double  its 
tenacity,  and  about  f  its  crushing  resistance  ;  and  of  copper  it 
is  about  f  its  tenacity. 

The  following  table,  which  gives  the  results  of  some  experi- 
ments upon  punching  plate  iron,  illustrates  the  law  of  resistance, 
and  gives  the  value  of  Ss  for  that  material. 

*  Weisbach  Mech.  and  Eng.,  vol.  i.,  p.  407. 

f  Kirkaldy's  Exp.  Inq.,  p.  71. 

\  Proc.  Inst.  Mech.  Eng.  England,  1858,  p.  76. 

§  Proc.  Inst.  Mech.  Eng.  England,  1858,  p.  73. 

|  Stoney  on  Strains,  vol.  ii.,  p.  284. 

•jf  Barlow  on  the  Strength  of  Materials,  p.  24. 

**  Engineering  Statics,  Gillespie,  p.  33. 

ff  Tredgold's  Carpentry,  p.  42. 

ft  Murray  on  Shipbuilding  Wood  and  Iron,  p.  94. 


SHEARING    STRESS. 


91 


TABLE 

Of  Experiments  on  Punching  Plate  Iron. ' 


Diameter  of  the 

Thickness  of  the 

Sectional  area  cut 

Total  pressure  on 

Pressure  per  square 

hole. 

plate. 

through. 

the  punch. 

inch  of  area. 

Inch. 

Inch. 

Square  inch. 

Tons. 

Tons. 

0-259 

0-437 

0-344 

8-384 

24-4 

0-500 

0-625 

0-982 

26-678 

27-2 

0-750 

0-625 

1-472 

34-768 

23-6 

0-875 

0-875 

2-405 

55-500 

23-1 

1-000 

1-000 

3-142 

77-170 

24-6 

These  results  give  for  the  value  of  Ss  from  51,000  Ibs.  to 
61,000  Ibs.  The  total  resistance  varies  nearly  as  the  cylindrical 
surface  of  the  hole. 

APPLICATIONS. 

81.  PROBLEM  OF  A  TiK-KEAui. — To  find  the  relation  be- 
tween the  distance  AB,  Fig.  32,  and  the  depth  of  a  rectangu^ 
lar  beam  below  the  notch,  so  that  the  total  shearing  strength 
shall  equal  the  total  tenacity. 


If * 


FIG.  32. 


Let  h  —  AB  =  the  distance  of  the  bottom  of  the  notch  from  the 

end, 

d  =  the  remaining  depth  of  the  beam, 
k  —  the  section  of  AB, 
K  =  the  section  belowfA?\ 
T  =  the  modulus  of  tenacity,  and 
/Ss  =  the  modulus  of  shearing  strength : 
Then  the  condition  requires  that 

TK  =  &A, but  *:£::*:<* 


__ 
•*  K  ~  d  ~  ~Ss 


*  Proceedings  Inst  Mech.  Eng.,  1858,  p.  76. 


•Z.GL  X 


v.fl  o. 

92  THE   SESISIAFCE   OF   MATERIALS.   V  ' 

(h    n       r*,  ff 

Wfr   0 1  ;     /  T       12000 

Example. — For  Oak  ^-  =  =15^  nearly;  hence  AB  should  be  about  15£ 

O$  9  O\J 

times  the  remaining  depth. 

8  3.   RIVETED  PLATES. — Given  the  diameter  of  the  rivets  ; 
it  is  required  to  find  the  distance  between  them  from  centre  to 
centre,  so  that  the  strength  of  the  rivets  for  a  single  row  shall 
equal  the  strength  of  the  remaining  iron  in  the  plates. 
Let  d  =  the  diameter  of  the  rivets, 

c  —  the  distance  between  them  from  centre  to  centre, 

Jc  —  the  section  of  the  rivet, 

K  =  the  remaining  section  of  the  plate,  and 

t  =  the  thickness  of  the  plate. 
For  iron  T  =  /Ss /    hence,  proceeding  as  above,  and  we  have 


Examples. — If  t  —  J  inch,  and  d  —  $  inch 
thenc  =  1.2854,  inch, 


and 


-0.61.  ~   <^2 


If  t  =.  i  inch,  and  d  =  f  inch  ;  then  c  =  0.8238  and  -   —  =  0.544,  which  is 

c 

nearly  the  value  given  by  Fairbairn  for  the  strength  of  single  riveted  plates. 
See  Article  27.  To  insure  this  strength  the  rivet  should  fit  tightly  in  the  hole. 

83.   LONGITUDINAL  SHEARING  IN  A  BENT  BEA3E. When 

a  beam  is  subjected  to  a  transverse  stress,  we  have  already  seen, 
Articles  74  and  75,  that  the  fibres  are  unequally  strained, 
and  hence  are  unequally  elongated  and  compressed.  This  can- 
not be  done  without  producing  a  shearing  stress  between  the 
adjacent  elements  or  fibres,  as  shown  in  Figs.  27  and  28.  This 
shearing  strain  rarely  overcomes  the  cohesion  of  the  particles, 
but  if  they  were  held  only  by  friction  it  might  overcome  that. 
To  illustrate  this  latter  idea,  suppose  several  boards  from 
ordinary  lumber  are  placed  upon  each  other,  and  the  whole 
supported  at  the  ends  in  any  convenient  way.  When  in  this 
condition  draw  several  straight  lines  across  the  pile,  perpen- 
dicular to  the  central  board.  Then  deflect  the  whole  by  a 
weight  at  the  middle,  or  in  any  other  convenient  manner,  and  it 
will  be  observed  that  the  lines  are  no  longer  straight,  but  bro- 
ken, and  the  general  direction  does  not  remain  normal  to  the 
axis  or  central  board.  In  the  experiment  the  top  layer,  instead 

N^(_.      ^~  t**'  -    GA**-t~4  t/v*-*~'  nrQs/L^o    oj.    {/\/^t^  0*iML  4  *•*•*  &>k 

e       °^n        -Td   4-  ft  & 

&  fr-  fa^A.  (UL^  .-  i.  &L 

v  - 

'^KA^l^  -C, 


SHEARING    STRESS.  93 

of  being  shortened  as  in  a  solid  beam,  retains  its  length  by 
overcoming  the  friction  between  the  top  board  and  the  one  im- 
mediately under  it.  The  friction,  whether  it  be  much  or  little, 
represents  the  shearing  stress  in  a  beam. 

The  elongations  and  compressions  of  the  fibres  in  a  bent 
beam  being  proportional  to  their  distances  from  the  neutral 
axis.  Article  74,  it  follows  that  the  shearing  stress  is  evenly  dis-        l^. 
tributed   over  the   cross  section  ;    and  that,  beginning  at  they/ 
lixis,  the  total  shearing  stress  increases  uniformly  with  the  dis- 
tance from  the  axis.      In  a  beam  which  is  bent  by  forces  per- 
pendicular to  the  axis  the  shearing  resistances  to  compression 
and  tension  form  a  couple  whose  arm  is  the  distance  between.  A,/ 
the  centre  of   the   compressed  section  and  the  centre  of  the 
extended  section.      This  resistance  in  bent  beams  is  generally^' 
elastic.      The  coefficient  of  elasticity  for  this  case  for  fibrous^ 
bodies  has  not  been  determined. 

84:.    TRANSVERSE     SHEARING     IN    BENT      BEAMS. Quite 

analogous  to  the  preceding  case  is  that  of  transverse  shearing  in 
a  beam  which  is  bent  by  external  forces.  Referring  to  Fig.  28, 
in  order  that  the  weight  P  should  be  sustained  by  the  horizon- 
tal beam,  there  is  necessarily  a  vertical  force,  or  a  vertical  com- 
ponent of  forces  in  the  beam,  and  it  is  the  same  at  all  sections 
between  A  and  B.  This  is  easily  shown  by  the  principles  of  -X 
mechanics. 

In  order  to  simplify  the  problem,  suppose  that  all  the  bend- 
ing forces  are  in  a  plane,  and  let 
P,  P1?  P2,  &c.,  be  the  bending  forces, 

F,  F1?  Fa,  &c.,  be  the  forces  in  a  beam,  each  of  which  is  the  re- 
sultant of  all  the  forces  concurring  at  that  point, 
«,  "o  *a>  &c-> tne  angles  which  P,  Pa,  &c.,  make  with  the  axis 

of  x, 
a,  «u  #«  &c.,  the  angles  which  F,  F,,  Fa,  &c.,  make  with  the 

axis  of  x,  and  y  an  axis  perpendicular  to  x. 
Then  the  principles  of  statics  give  the  following  equations : 
sPcosa  +  sFcosa  =  0, 

~n      •  r\ 

sP  sm  *  +  sF  sin  a  =  0, 

s(Py  cos  *  —  P#  sin  «)  +  s  (Fy  cos  a  —  F#  sin  a)  =  0. 
Let  x  coincide  with  the  axis  of  the  beam,  and  let  all  the  forces 

be  vertical;  or  *  =  90°  or^feO0  ;  then 

>  i  /^s. 

-  faUffyfau*^^  *<*  &**/vf 

i  *~l^UtulL  d&esR^ 
^f^^ft^ 


94  THE  RESISTANCE   OF   MATERIALS. 

'    (1)    -    -    -    sF  cos  a  =  0 

(2)  -  s±P+sFsina  =  0 

(3)  -    -     .     s  ±  Pa?  +  sFy  cosa  —  sF#  sin  a  =  0 

The  first  of  these  equations  shows  that  the  sum  of  the  resist- 
\^  ing  forces  parallel  to  the  axis  is  zero  ;  or  that  the  total  compres- 
sion equals  the  total  tension.  This  is  equation  (35)  in  another 
form.  The  second  shows  that  the  sum  of  the  bending  forces 
equals  the  sum  of  the  vertical  components  of  the  resisting  forces. 
If  we  let  8s  represent  a  strain  as  well  as  a  modulus,  this  equa- 
tion becomes  sP  =  sF  sin  a  —  $?,  which  is  the  result  sought. 

This  is  as  far  as  it  is  necessary  to  carry  the  investigation  in 
this  connection  ;  but  it  may  be  well  to  show  the  use  of  the 
third  equation.  If  we  use  a  resultant  moment  for  each  of  the 
above  sets  of  moments  the  equation  becomes 

P  V  —  x"  sF  sin  a  =  Fy  ,     CxH  GL 
or,  PV  -  x"  Ss  =  Fy  ;  but  Ss  =  sP  =  P', 
.-.  P'  (a,'  _  x")  =  Fy  ; 

hence  the  shearing  stress  forms  a  couple'  with  the  applied  force, 
—  or  resultant  of  applied  forces.  This  equation  under  the 
form  , 


is  an  essential  one  in  Articles  86  and  136. 

Examples  of  transverse  shearing  stress.     The  second  of  equations  (44a),  as 
reduced  is, 

Ss  =  LP. 

1.  Let  a  beam  be  uniformly  loaded  over  its  whole  length,  and  supported  at 
its  ends  as  in  Fig.  42, 

and  let    w  =  the  load  on  a  foot  of  length. 
I   =  the  length  of  the  beam, 

V  =  %wl  =  the  amount  sustained  at  each  support, 
x  =  any  distance  from  either  end-;  then 

wx  =  the  load  on  the  length  x  ;  and  the  expression  for  the  shearing 
stress  becomes 

Ss  =  iwl  —  wx, 

which  is  the  equation  of  a  straight  line  (see  Fig.  100).    Its  value  is  greatest  for 
x  =  0,  for  which  it  is  but  —  i  W  ;  and  is  zero  for  x  —  \l. 

2.  Suppose  the  beam  is  supported  at  its  ends,  and  has  a  weight  at  the 
middle  of  its  length. 

Let  P  =  the  weight,  and  the  other  notation  as  before  ;  then  V  =  iP,  and 
s  —  iP  —  0  to  the  middle,  and  beyond  the  middle  8s  =  i?  —  P  =  —  iP  ;  and 
hence  it  is  constant  over  its  whole  length. 


: 


SHEARING    STRESS.  95 

3.  If  the  beam  sustains  a  uniform  load,  and  also  a  uniformly  increasing 
load  from  one  end,  as  in  Fig.  98,  in  which  Wi  is  the  total  load  which 
increases ; 

we  have  8s  -  V  -  wx  -  W»£  '  (Ar  C   .'  ,   $5   ~A~-Ur  I 


85.  SHEARING  RESISTANCE  TO  TORSION. — When  a  piece  is 
twisted  there  is  a  tendency  in  one  section  to  slip  over  the 
adjacent  one,  and  the  corresponding  resistance  constitutes 
a  shearing  strain.  It  is  least  at  the  axis,  and  increases  gradu- 
ally as  we  proceed  from  it. 

I  * 


£> 

*"  ^  (      0*+*-       T7&-  ' 

f   t^     .^mmw  £/V      t      j 


r 

•        .  ra^c-'- 

.  ' 
^/»^fa~M^?  UM 


/ 

^v^*-7 


Ut*   t 

I  k^j4f~i,4  <Y,-  t  /^ 


96  FLEXURE. 


CHAPTER    Y. 

FLEXURE. 
ELASTIC    CURVE. 

WHEN  a  beam  is  bent  by  a  transverse  strain,  equilibrium  is 
established  between  the  external  and  internal  forces  ;  or,  to  be 
more  specific,  all  the  external  forces  to  the  right  or  left  of  any 
transverse  section  are  held  in 
equilibrium  by  the   elastic 
resistances  of  the  material 
in  the   section.     When   in 
this  state  the  curve  assumed 
by  the  neutral  axis  is  called 
the  elastic  curve. 

To  find  the  general  equa- 
tion of  the  elastic  curve,  let 

JblG.  oo. 

Fig.  33  represent   a  beam, 

fixed  at  one  end,  or  supported  in  any  manner,  and  deflected 
by  a  weight,  P,  or  by  any  number  of  forces.  AB  is  the 
neutral  axis.  Take  the  origin  of  coordinates  at  B  (or  at  anj 
other  point  on  the  neutral  axis),  and  let  x  be  horizontal  and 
coincide  with  the  axis  of  the  beam  before  flexure,  y  vertical 
and  u  perpendicular  to  the  plane  of  xy.  The  transverse  sections 
CM  and  EF  being  consecutive  and  parallel  before  flexure,  will 
meet  after  flexure,  if  sufficiently  prolonged  in  some  point,  as 
o.  Through  N  draw  KII  parallel  to  CMy/hen  will  ke  be  the 
elongation  of  a  fibre  whose  original  length  was  ck.  We  have 
the  following  notation : — * 

dx  =  LN"  —  the  distance  between  consecutive  sections, 

y  —  N0  =  any  ordinate  of  the  surface, 

u  —  ~Na  or  "No,', 

~b  =  1SN'—  the  limiting  value  of  u, 
f  (y>u}  —  equation  of  the  transverse  section, 

*  Several  of  the  more  important  problems  of  this  chapter  are  solved  in  Arti- 
cles 93  to  103,  without  the  use  of  the  calculus. 


FLEXURE.  97 


dy  du  —  the  transverse  section  of  a  fibre, 
limiting  valtae  of  u,,« 


p  =  ON  —  the  radius  of  curvature  at  IN", 
p  =  the  force  necessary  to  elongate  any  fibre  an  amount  equal 
to  A  when  applied  in  the  direction  of  its  length, 
A  =  ke, 

1  —  the  moment  of  inertia  of  the  section, 

E  =  the  coefficient  of  elasticity  of  the  material,  which  is  sup- 
posed to  be  the  same  for  extension  and  compression, 

2  P#  =  a   general   expression  for  the   moment    of  applied 
forces. 

We   suppose  that  the  strain  is  within  the  elastic  limit,  and 


establish  the  algebraic  equation  on  the  condition  that  the  sum  of 
the  moments  of  the  applied  or  deflecting  forces  equals  the  sum  of  f 
the  moments  of  the  resisting  forces.     We  also  assume  that  the 
neutral  axis  coincides  with  the  centre  of  the  transverse  sections 
of  the  beam. 

By  the  similarity  of  the  triangles  LOIST  and  £N^,  we  have 

ON  :  Ne  : :  LIST :  Jce,  or  p :  y  : :  dx  :  A 


f 

The  force  necessary  to  produce  this  elongation  is  (see  equa- 
tion (3)), 

A 

dx* 
which  becomes,  by  substituting  A  from  (45), 

p  =  —ydydu (46) 

P 
and  the  moment  of  this  force  is  found  by  multiplying  it  by  y ; 

.'.  py  —  -•  y'  dy  du  (47) 

The  total  moment  of  all  the  resisting  forces  to  extension  and 
compression  is  found  by  integrating  (47)  so  as  to  include  the 
whole  transverse  section,  and  this  will  equal  the  sum  of  the 
moments  of  the  applied  forces : 

I"      S*b      /*  +  ^  f*b      /*Q  "I 

/        /       frdydu  +    I       /    tfdyfa      = 
(    L^O     Jo  t/0     •/-«  -J   ' 


E  r  r+u  - 

t         i         y*dy  du 

9  Jo   /-* 


THE   RESISTANCE   OF  MATERIALS. 

(XT   —       f  i  tfrlfii  AM.  —  "sP^y?       _  _____      fAQ\ 

\±0) 

The  quantity  'EJJy'dy  du,  which  depends  upon  the  form  of 

the  transverse  section  and  nature  of  the  material,  is  called  the 
moment  of  flexure. 

The  quantity  f  I  y*dy  du,  when  taken  between  limits  so  as  to 

include  the  whole  transverse  section,  is  called  the  moment  of 
inertia  of  the  surface.*  Calling  this  I  and  equation  (48) 
becomes 

El 

7  =  *p* w 

which  is  the  equation  of  the  elastic  curve. 

An  exact  solution  of  equation  (49)  is  not  easily  obtained  in 
practice,  except  in  a  few  very  simple  cases ;  but  when  the  deflec- 
tion is  small  an  approximate  solution,  which  is  generally  com- 
paratively simple  and  always  sufficiently  exact,  is  easily  found. 

I 


&y  dx  d?y 

dx* 
=  -35-  nearly,  since  for  small  deflections 

-~  (which  is  the  tangent  of  the  angle  which  the  tangent  line, to 
dx 

the  curve  makes  with  the  axis  of  x)  is  small  compared  with 
unity,  and  hence  may  be  omitted.    Hence  equation  (49)  becomes 


(50) 

UWp 

which  is  the  general  approximate  equation  of  the  neutral  axis. 

87.  THE  MOMENT  OF  INERTIA  t  of  all  transverse  sec- 
tions of  a  prismatic  beam,  is  constant,  and  hence  I  is  constant 
for  prismatic  beams. 

*  See  Appendix.  f  See  Appendix. 


t 


FLEXURE. 

For  a  rectangle,  as  Fig.  34,  we  have 
I  =     /          /        y^dydu  =  -fa  bd?    -    (51)   ~ 


Jo     J-\A 


For  a  circle,  the  origin  of  coordinates  being  at 
the  centre ; 


dydu  =  rdrdt 


/V      /~2?r 
.-.!=    /         /        r'drd* 

Jo    Jo 


FIG.  34. 


FIG.  35. 


-    -    (52) 


SPECIAL  CASES  OF  PRISMATIC  BEAAIS. 
88«  REQUIRED  THE  EQUATION  OF  THE  NEUTKAL  AXIS,  AMOUNT 

OF  DEFLECTION,  AND  SLOPE  OF  THE  CURVE  OF  A  PRISMATIC  BEAM, 
WHEN  SLIGHTLY  DEFLECTED,  AND  SUBJECTED  TO  CERTAIN  CONDITIONS 
AS  FOLLOWS  I 

89.  CASK  i. — SUPPOSE  A  HORIZONTAL  BEAM  is  FIXED  AT  ONE 

EXTREMITY  AND  A  WEIGHT  P  RESTS  UPON  THE  FREE  EXTREMITY; 
REQUIRED  THE  EQUATION  OF  THE  NEUTRAL  AXIS  AND  THE  TOTAL 
DEFLECTION. 


Fio.  37. 


FIG.  36. 


The  beam  may  be  fixed  by  being  imbedded  firmly  in  a  wall,  as 
in  Fig.  36,  or  by  resting  on  a  fulcrum  and  having  a  weight  ap- 
plied on  the  extended  part,  which  is  just  sufficient  to  make  the 
curve  horizontal  over  the  support,  as  in  Fig.  37.  The  latter 

- 


100  THE   RESISTANCE    OF    MATERIALS. 

case  more  nearly  realizes  the  mathematical  condition  of  fixed- 
ness. In  either  case  let 

I  =  AB  —  the  length  of  the  part  considered, 
i  —  the  inclination  of  the  curve  at  any  point,  and 
A  =  BC  =  the  total  deflection. 

Take  the  origin  of  coordinates  at  the  free  end,  A  ;  x  horizontal, 
y  vertical  and  positive  downwards.  •  The  moment  of  P  on  any 
section  distant  x  from  A  is  P#,  which  is  the  second  member  of 
equation  (50)  in  this  case.  Hence  the  equation  becomes 


Multiply  both  members  by  the  dx  and  integrate,  and  we  have 
EljI^Ptf  +  C,     ........  (54) 

When  the  deflections  are  small,  the  length  of  the  beam  re- 
mains sensibly  constant,  hence  for  the  point  B,cc  —  /;   and  at 

fj/ti 
the  fixed  end  -y-  =  0.     Substitute  these  values  in  equation  (54), 

and  we  find  C,  =  —  ^  PZ2,  and  (54)  gives 

.....      (55) 


The  integral  of  equation  (55)  is 


But  the  problem  gives  y  =  0  for  x  =  0  .*.  C2  =  0  ; 

(56) 


which  is  the  equation  of  the  neutral  axis,  and  may  be  discussed 
like  any  other  algebraic  curve. 

The  greatest  slope  is  at  A,  to  find  which  make  x  —  0  in  equa- 
tion (55) 

/.  tang  i  (at  the  free  end)  =  -  ^L 
} 


<•*—       £&z 

The  greatest  distance  between  the  curve  and  the  axis  of  x  is 
at  B,  to  find  which  make  x  =  I  in  equation  (56),  and  we  have 

fe; 


FLEXURE.  101 

If  y  were  positive  upward,  everything  else  remaining  the 
same,  the  second  member  of  equation  (53)  would  have  been 
negative,  for  it  is  a  principle  in  the  differential  calculus  that 
when  the  curve  is  concave  to  the  axis  of  x,  the  second  differen- 
tial coefficient  and  the  ordinate  must  have  contrary  signs.  This 
would  make  tang  i  and  A  positive.  It  will  be  a  good  exercise  for 
the  student  to  solve  this  and  other  problems  by  taking  the  origin 
of  coordinates  at  different  points,  only  keeping  x  horizontal  and 
y  vertical.  For  instance,  take  the  origin  at  B  ;  at  C  ;  at  the 
point  where  the  free  end  of  the  beam  was  before  deflection  ;  at 
the  middle  of  the  beam  ;  or  at  any  other  point. 

Example.—  If  I  =  5  ft.,  t>  =  3  in.,  d  =  8  in.*,  E  =  1,600,000  Ibs.,  and  P  =  5,000 
Ibs.  ;  required  the  slope  at  the  free  end  and  at  the  middle,  and  the  maximum 
deflection,X  ^<^  f(  i  —  /  f  —  ^  Z  / 


CASE  II.  -  SUPPOSE  THAT  THE  BEAM  IS  FIXED  AT  ONE  END, 
IS   FREE  AT   THE   OTHER,  AND   HAS   A  LOAD  UNIFORMLY   DISTRIBUTED 

OVER  ITS  WHOLE  LENGTH.  —  The  beam  may  be  fixed  as  before,  as 
shown  in  Figs.  38  and  39. 


FIG.  33. 

Fro. 

Let  w  =  the  load  on  a  unit  of  length.     This  load  may  be  the 
weight  of  the  beam,  or  it  may  be  an  additional  load. 
W  —  wl  =.the  total  load. 
Take  the  origin  at  A. 

Then  wx  =  the  load  on  a  distance  a?,  and 

4mBa=:the  moment  of  this  load  on  a  section  distant 

x  from  A. 
Hence  equation  (50)  becomes 


dy       w 
'        = 


W 


102  THE   RESISTANCE   OF   MATERIALS. 

WV  WT 

an*A  =   -         =    -  ......    -    (61) 


y  =  0  for  x  =  0  /.  C\  =  0,  and 
y  ==  A  for  x  =  L 

If  the  origin  of  coordinates  were  at  the  fixed  end,  s  P#  in  the 


w 
first  case  would  be  P  (I  —  #),  and  in  the  second  ^  (Z  —  »)*•   The 

student  may  reduce  these  cases  and  find  the  constants  of  inte- 
gration. This  case  may  be  further  modified  for  practice  by 
taking  the  origin  of  coordinates  at-  different  points. 

91.  CASE  in.  —  LET  THE  BEAM  BE  FIXED  AT  ONE  END  AND  A 

LOAD    UNIFORMLY   DISTRIBUTED    OVER    ITS    WHOLE    LENGTH,    AND    A 

WEIGHT  ALSO  APPLIED  AT  THE  FREE  END.  —  This  is  a  combination 
of  the  two  preceding  cases,  and  is  represented  by  Figs.  36  and 
37,  in  which  the  weight  of  the  beam  is  the  uniform  load. 


/.El        = 


(62) 


hence  the  deflection  of  a  beam  fixed  at  one  end  and  free  at  the 
other,  and  uniformly  loaded,  is  f  as  much  as  for  the  same  weight 
applied  at  the  free  end. 


.  CASE  iv.  —  LET  THE  BEAM  BE  SUPPORTED  AT  ITS  ENDS 
AND  A  WEIGHT  APPLIED  AT  ANY  POINT.  —  Figs.  4:0  and  41  represent 
the  case. 


Fis.  40. 

FIG.  41. 

Let  the  reaction  of  the  supports  be  Y  and  Y4.    Take  the 


FLEXURE.  103 

origin  at  A  over  the  support,  and  let  AD  =  c  =  the  abscissa 
of  the  point  of  application  of  P. 

Then,  Y  =~  P,  and  Y,  =  j  P. 
t>  & 

The  case  is  the  same  as  if  a  beam  rested  on  a  support  at  D, 
and  weights  equal  to  Y  and  Y!  were  suspended  at  the  ends. 
For  the  part  AD,  equation  (50)  becomes  :  — 


(63) 


x7?y  P(7  _  A 

•   T^-^KT^  «•  +  €!,; (64) 

dx  2ZEI 

T3/7  W\ 

-C^  +  ^^rO);   -    -    -     (65) 

VSC/J-^-i. 

in  the  last  of  which,  y  =  0  f  or  a?  =  0  /.  Ca  =  0  as  indicated. 

For  the  part  DB,  the  origin  of  coordinates  remaining  at  A, 
we  have : — 


and      =(W-6W)  +  C'x  +  C".    -    -    -    -    (68) 


To  find  the  constants,  make  x  =  c  in  equations  (64)  and  (67) 
and  place  them  equal  to  each  other  ;  do  the  same  with  (65)  and 
(68)  ;  and  also  observe  that  in  (68)  y  =  0  for  x  =  I.  These  con- 
ditions establish  the  three  following  equations  :  — 


104  THE   KESISTANCE   OF    MATEEIALS. 

From  these  we  find 


r\/f __ 


6EI 


Hence,  for  the  part  AD  we  have 


dy 
tix 


or^--^- 
01'^~6EU 


6EK 


(69) 


(70) 


To  find  the  maximum  deflection,  if  c  is  greater  than  JZ,  make 
^x  =  0  in  (69)  and  find  x ;  then  substitute  the  value  thus  found  in 

equation  (70).  If  c  <-JZmake-r-  —  0  inequation  (67)  and  substi- 
tute the  value  thus  found  in  equation  (68). 

If  D  is  at  the  middle  of  the  length,  make  o  =  iZ  in  equations 
(63),  (69),  and  (70) ;  and  we  have  for  the  curve  AD 

=  -iPa, (71) 


and  A  =  -  (if  *  =  i?  in  (72) ) 


(72) 

(73) 

The  greatest  stress  is  at  the  centre,  and  the  maximum  mo- 
ment is  found  by  making  OB  —  \l  in  the  second  member  of  equa- 
tion (71).  Hence,  the  maximum  moment  is 


FLEXURE. 


105 


In  this  case  the  curve  DB  is  of  the  same  form  as  AD,  but  its 
equation  will  not  be  of  the  same  form  unless  the  origin  of  co- 
ordinates be  taken  at  the  other  extremity  of  the  beam. 

93.    CASE  v.  —  SUPPOSE   THAT  A  BEAM  is   SUPPORTED  AT  OR 

NEAR   ITS   EXTREMITIES,  AND    THAT    A  LOAD    IS  UNIFORMLY  DISTRIB- 
UTED   OVER   ITS  WHOLE   LENGTH. 


No  account  is  made  of  the  small  por- 
tion of  the  beam  (if  any)  which  projects 
beyond  the  supports.  The  distance  be- 
tween the  supports  is  the  length  of  the 
beam  which  is  considered. 

Let  the  notation  be  the  same  as  in  the 
preceding  cases. 


42- 


Then  Y=  \wl  =  ^W=  the  weight  sustained  by  each  support  ; 
~Vx  =  %wlx  =  the  moment  of  Y  on  any  section,  as  c  ; 
wx  is  the  load  on  x,  and  the  lever  arm  of  this  load  is 
the  distance  from  its  centre  to  the  section  c,  or  ^x  ;  hence  its 
moment  is  ^m#2,  and  the  total  moment  is  the  difference  of  the 
two  moments.     Hence  ecpation  (50)  becomes 

Ei=M-&+ao;  —  (74) 


=_.    -    (76)     ^ 

In  these  equations  ^=0  for  x  =  $,  .'.  C,  =  -—  ; 

and  y  =  0  for  x  =  0,  /.  <72  =  0. 
94.    CASE  vi.  —  LET  THE   BEAM  BE  SUPPORTED  AT  ITS  ENDS, 

UNIFORMLY    LOADED,    AND    ALSO     A    LOAD    MIDWAY   BETWEEN   THE 
SUPPORTS. 

This  case  is  a  combination  of  the  two  preceding  ones,  and 


106  THE   RESISTANCE   OF   MATERIALS. 

may  be  represented  by  Fig.  40  ;  for  the  weight  of  the  beam 
may  be  the,  uniform  load.     Hence, 

-    -    -         (77) 


Experiments  on  the  deflection  of  beams  are  generally  made 
in  accordance  with  this  case.  If  the  beam  be  rectangular,  we 
have  from  equation  (51), 


which  in  (78)  gives 


In  making  an  experiment  to  determine  E,  the  beam  is  weighed, 
and  that  portion  of  it  which  is  between  the  supports  and  unbal- 
anced will  be  "W,  and  all  the  quantities  except  E  may  be  directly 
measured.  If  E  be  known,  we  may  measure  or  assume  all  but 
one  of  the  remaining  quantities,  and  solve  the  equation  to  find 
the  remaining  quantity,  as  the  following  examples  will  illus- 
trate :  — 


Examples.  —  1.  If  a  rectangular  beam,  5  feet  long,  3  inches  wide,  and 
3  inches  deep,  is  deflected  -fe  of  an  inch  by  a  weight  of  3,000  Ibs.  applied  at  the 
middle  ;  required  the  coefficient  of  elasticity.  E  =  20,000,000. 

2.  If  b  =  2  inches,  d  =  4  inches,  and  I  =  6  feet,  the  weight  of  the  beam  144 
Ibs.,  and  a  weight  P=10,000  Ibs.  placed  at  the  middle  of  the  beam  deflects  it  i 

^j    an  inch  ;  required  B.  /  Y;  7//,  i-  1  0.  E  =  14,580,000  Ibs. 

3.  A  joist,  whose  length  is  16  feet,  breadth  2  inches,  depth  12  inches,  and  co- 
efficient of  elasticity  1,600,000  Ibs.,  is  deflected  £  inch  by  a  weight  in  the 
middle  ;  required  the  weight  ;  the  weight  of  the  beam  being  neglected. 

Ans.  P  =  1,562  Ibs. 

4.  An  iron  rectangular  beam,  whose  length  is  12  feet,  breadth  1£  inches,  co- 
efficient of  elasticity  24,000,000  Ibs.,  has  a  weight  of  10,000  Ibs.  suspended  at 
the  middle  ;  required  its  depth  that  the  deflection  may  be  4^  of  its  length. 

Ans.  8.8  in. 

5.  A  rectangular  wooden  beam,  6  inches  wide  and  30  feet  long,  is  supported 
at  its  ends.     The  coefficient  of  elasticity  is  1,800,000  Ibs.  ;  the  weight  of  a 


FLEXURE.  1  07 

cubic  foot  of  the  beam  is  50  Ibs.  ;  required  the  depth  that  it  may  deflect  1  inch 
from  its  own  weight.  fc>  ^  >is*st&AA  -J 

How  deep  must  it  be  to  deflect  T£  „  of  its  length  ?    £  J  fr~  </u-d/&~* 
6.  A  cylindrical  beam,  whose  diameter  is  2  inches,  length  5  feet,  weight  of  a 
cubic  inch  of  the  material  0.25  lb.,  is  deflected  $  of  an  inch  by  a  weight  P  = 
3,000  Ibs.  suspended  at  the  middle  of  the  beam.     Bequired  the  coefficient  of 
elasticity.  Jff 

To  solve  this  substitute  I  =  far*  (equation  (52))  in  equation  (8.  This  gives 


7.  Required  the  depth  of  a  rectangular  beam  which  is  supported  at  its  ends, 
and  so  loaded  at  the  middle  that  the  elongation  of  the  lowest  fibre  shall  equal 
T^riJ  of  its  original  length.  (Good  iron  may  safely  be  elongated  this  amount.) 

Equations  (49)  and  (73a)  become  —  =  ±Pl.'.  p=*5.     In  this  substitute  the 

P  PI 

value  of  I,  equation  (51),  and  it  becomes 


3PZ 


=  700e* 


2100K 


8.  Required  the  radius  of  curvature  at  the  middle  point  of  a  wooden  beam, 
when  P  =  3,000  Ibs.  ;   1  =  10  ft.  ;  b  =  4  in.;   d=:8in.;  and  B  =  1,000,000  Ibs. 

^^=  ,,806  inches. 

9.  Let  the  beam  be  iron,  supported  at  its  ends.     Let  b  =  1  in.  ,  d  =  2  in.,  Z  = 
8  f  t.  ,  E  =  20,000,000  Ibs.     Required  the  radius  of  curvature  at  the  middle  when 
the  deflection  is  i  of  an  inch.     Use  eqs.  (49)  and  (73)  for  P  at  the  middle. 


4 

from  which  it  appears  that  it  is  independent  of  the  breadth  and  depth. 

10.  The  centrifugal  force  caused  by  a  load  moving  over  a  deflected  beam 

may  be  found  from  the  expression  ™—  ,  hi  which  m  is  the  mass  of  the  moving 

9 

load,  r>  its  velocity  in  feet  per  second,  and  p  the  radius  of  curvature  of  the 
beam.     (See  Mechanics.) 

11.  All  these  problems  may  be  applied  to  beams  fixed  at  one  end,  and  P  ap- 
plied at  the  .free  end,  or  for  a  load  uniformly  distributed  over  the  whole  length, 
by  using  the  equations  under  Cases  I.  ,  II.  ,  and  III. 

According  to  equation  (79)  the  deflection  varies  as  the  cube 
of  the  length  ;  and  inversely  as  the  breadth  and  cube  of  the 

depth,  and  directly  as  the  weight  applied.     * 

- 


108  THE   RESISTANCE   OF   MATERIALS. 

96.  BARLOWS  THEORY  has  not,  to  my  knowledge,  been 
applied  to  flexure,  but  it  may  be  well  to  inquire  what  effect  it 
would  have.  In  the  common  theory,  it  is  assumed  that  the  total 
force  is  expended  in  elongating  and  compressing  the  fibres  ; 
but,  according  to  Barlow's  theory,  a  portion  of  the  force  is 
absorbed  in  drawing  (so  to  speak)  one  fibre  over  the  adjacent 
one  ;  hence  the  deflection  should  be  less  by  this  theory  than  by 
the  common  one. 

An  experiment  made  by  Mr.  Hatcher,  England,  showed 
that  it  was  less.  (See  Mosley's  Mechanics  and  Engineering,  p. 
514.) 

To  find  E  by  this  theory,  $  will  represent  &  fractional  part  of 
the  strain  (not  of  the  ultimate  resistance). 

Then  $  ify  dy  dx  is  the  moment  of  resistance  to  longitudinal 

shearing. 

Hence  we  have 


Or  for  a  rectangular  beam  supported  at  the  ends,  we  have, 
by  combining  the  general  moments  of  equations  (71)  and  (74:)  and 
using  y  positive  upwards  :  — 

(81) 


4>  is  very  small  for  small  deflections,  but,  whatever  its  value, 
we  see  that  E  found  by  this  method  will  be  less  than  that  found 
by  the  common  theory  ;  and  hence  less  than  that  given  by  the 
method  in  Article  7. 

97.  CASE  VII.  LET  THE*  BEAM  BE  FIXED  AT  ONE  EXTREMITY, 
SUPPORTED  AT  THE  OTHER,  AND  HAVE  A  WEIGHT,  P,  APPLIED  AT 
ANY  POINT. 


FIG.  44. 


FLEXURE.  109 

The  beam  may  be  fixed  by  being  encased  in  a  wall,  Fig.  43, 
or  by  extending  it  over  a  support  and  suspending  a  weight  on 
the  extended  part  sufficient  to  make  the  beam  horizontal  over 
the  support,  Fig.  44 ;  or  by  resting  a  beam  whose  length  is  2£ 
on  three  equidistant  supports,  and  having  two  weights,  each 
equal  to  P,  resting  upon  it  at  equal  distances  from  the  central 


support,  Fig.  45.     In  the  latter  case  each  half  of  the  beam  fulfils 
the  condition  of  the  case. 

Let  I  =  AB,  Fig.  43,  be  the  part  considered, 
Y  =  the  reaction  of  the  support, 
nl  =  AD  =  the  abscissa  of  P,  and 
f  ==.  the  deflection  of  the  beam  at  D. 

Take  the  origin  at  A,  the  fixed  end.  We  may  consider  that  the 
curve  DB  is  caused  by  the  reaction  of  Y,  while  all  the  forces  at 
the  left  of  P  hold  the  beam  for  Y  to  produce  its  effect.  Similarly 
the  curve  AD  is  produced  by  the  reaction  Y  and  the  weight  P, 
while  all  the  forces  at  the  left  of  them  hold  the  beam.  In  all 
cases  we  may  consider  that  the  applied  forces  on  one  side  of  the 
transverse  section  are  'in  equilibrium^with  the  resisting  forces 
of  tension  and  compression  in  the  section.  It  is  well  also  to 
observe  that  the  origin  of  moments  is  at  the  centre  of  the  trans- 
verse section,  while  the  origin  of  coordinates  may  be  at  any 
point. 

//77 

For  the  curve  AD  we  have,  observing  that  -j-  =  0  for  x  =  0, 
and  y  —  0  for  x  =  0  :  —  / 


(82) 
(83) 


110  THE   RESISTANCE   OF   MATERIALS. 

For  the  point  D,  we  have,  by  making  x  =  nl, 

;    -    -    -    (85) 


y  =f=  B»T-G»'-K)V]         •  (86) 


For  the  curve  DB,  observe  that  --  =  tang«  for  x  =  nl,  and 


y  ^yfor  #  =  nl,  using  for  their  values  (85)  and  (86)  in  deter- 
mining the  constants  in  the  following  equations,  and  we  have  :  — 


(88) 


'P--.  -    -    -    -    (89) 

To  find  the  reaction  Y,  observe  that  y  —  0,  for  x  =  I  in  (89), 
and  we  obtain  :  — 

0  =  (3  -  ri)Pn*?  -  2VZ3  ; 
.-.V  =  K(3-n)P.      -    -    -    -    (90) 

By  substituting  this  value  of  Y  in  the  preceding  equations, 
they  become  completely  determined.  For  the  curve  AT)  we 
shall  have  :  — 


EI       -  =  P[»Z  -  x-  K(»  -  n)(l  -  «)]  ;  (91) 


y  =  jCftiW  -  2z*  -  «'(»  -  «)  (Slaf-af)']  ;  -    (93) 
and  for  the  curve  DB  :  — 


FLEXURE.  Ill 


ar)p-HK.   -     (96) 

The  points  of  greatest  strain  in  these  curves  are  where  the  sum 
of  the  moments  of  applied  forces  is  greatest,  and  this  is  greatest 
when  the  second  members  of  (91)  and  (94)  are  greatest.  Neither 
of  these  expressions  have  an  algebraic  maximum,  and  hence 
we  must  find  by  inspection  that  value  of  x  which  will  give  the 
greatest  value  of  the  function  within  the  limits  of  the  problem. 
Equation  (91)  has  two  such  values,  one  for  x  =  0,  the  other  for 
x  =  nl,  and  equation  (94)  has  one  such  for  x  =  nl,  which  value 
will  reduce  (91)  and  (94)  to  the  same  value. 

Making  x  =  0  in  (91)  gives  for  the  moment  of  maximum 
strain, 

sPaj  =  iPZ[2n-3nf+<|     ......    (97) 

For  the  moment  of  strain  at  P,  make  x  =  nl}  in  (91)  or  (94), 
and  we  have 

sP#  =  i  ?M  [-3  +  4n  -  O  .....     (98) 

To  find  where  P  must  be  applied  so  that  the  strain  at  the  point 
of  application  shall  be  greater  than  if  applied  at  any  other 
point,  we  must  find  the  maximum  of  (98)  :  — 

.•.Dn  =  0  =  -6tt  +  12n,f-4/i*  .....      (99) 

.-.  n  =  0.634  -h     ..........    (100) 

or  the  force  must  be  applied  at  more  than  -ffo  of  the  length  of 
the  beam  from  the  fixed  end.  This  value  of  n  in  (98)  gives, 

.174 


Equation  (99)  has  two  values  of  n,  but  the  other  is  not  within 
the  limits  of  the  problem. 

The  position  of  the  weight,  which  will  give  a  maximum  strain 
at  the  fixed  end,  is  found  by  making  (97)  a  maximum.  Pro- 
ceeding in  the  usual  way,  we  find  :  — 

"=l±i  1/3  =  0.422+        ......    (101) 

which  in  (97)  gives,  sPaj  =  PI  x  0.181       ....     (102) 

and  in  (98)  tPx  =  PZ  x  0.131  +  V 

To  find  where  P  must  be  applied  so  that  the  strain  at  the 


112  THE   RESISTANCE    OF   MATERIALS. 

point  of  application  will  equal  the  strain  at  the  fixed  end,  make 
equations  (97)  and  (98)  equal  to  each  other,  and  find  n.  This 
gives, 

n=\  3.  4141+;-     .......    -     (103) 

(  0.  5858  +. 

But  n  =  0.5858  +  is  the  only  practical  value. 

To  find  where  P  must  be  applied  so  that  the  curve  at  that 

point  shall  be  horizontal,  make  -j-  —  0,  and  x  =  nl  in  (95). 

(1. 
This  gives  n—  \  3.4141 

(  0.5858 

which  are  the  same  as  the  preceding  values  of  n.  To  find  the 
corresponding  deflection,  make  x  =  nl,  and  n  =  0.5858  +,  in 
(93),  and  we  find 

P73 
A  =  0.0098  ~    -     -    .......       (104) 

&L 

For   n  <  0.5858,  tang  i  is  +  ) 

n  >  0.5858,  tang  i  is  —  V        -     -     -     -     (105) 
^,=0.5858,  tang  i  is  Oj 

To  find  the  maximum  deflection  when  n  =  0.634.  make  ~-  =  0  in 

•  dx 

(92)  or  (95),  according  as  the  greater  deflection  is  to  the  right 
or  left  of  P.  But,  according  to  (105),  it  belongs  to  the  curve 
AD  ;  hence  use  (92).  Making  n  =  0.634  in  (92),  placing  it 
equal  zero,  and  solving  gives, 

x  =  0.  6045Z  ; 
which  in  (93)  gives, 

p?3 

y  =  A  =  0.00957  44--      -----     (106) 

Jiil 

To  find  where  P  must  be  applied  so  as  to  give  an  absolute 
maximum  deflection  ;  first  find  the  abscissa  of  the  point  of 
maximum  deflection,  when  P  is  applied  at  any  point  by  making 

=  0  in  (92),  and  thus  find 


which,  substituted  in  (93)  gives  the  corresponding  maximum 


FLEXURE.  113 

deflection.     Then  find  that  value  of  n  which  will  make  the 
expression  a  maximum. 

The  point   of  contra-flexure  in  the  curve  AD  is  found  by 

$?y 
making  -r-5—  0  in  (91)  (see  Dif.  Cal.)  which  gives, 

ClnJG 

_  Sin*  -  n*l  - 


The  second  member  of  (91)  is  the  moment  of  applied  forces, 
and  as  it  is  naught  at  the  point  of  contra-flexure,  it  follows  that 
at  that  point  there  is  no  bending  stress,  and  hence  no  elongation 
or  compression  of  the  fibres,  tut  only  a  transverse  shearing 
stress,  the  value  of  which  is  determined  in  Article  153. 

If  a  beam  rests  upon  three  horizontal  equidistant  supports, 
and  two  weights,  each  equal  P,  are  placed  upon  it,  one  on  each 
side  of  the  central  support  and  equidistant  from  it,  it  fulfills 
the  condition  of  a  beam  fixed  at  one  end  and  supported  at  the 
other,  as  before  stated,  and  the  amount  which  each  support  will 
sustain  for  incipient  flexure  may  easily  be  found  from  the  pre- 
ceding equations. 

The  three  supports  will  sustain  2P,  and  the  end  supports  each 
sustain  Y  =  %n\3  -  ri)P.  (See  Eq.  (90).) 

Hence,  the  central  support  sustains 

Y'  =  2l>-7i>(3-H)P. 

If  n  =  i,  Y  =  TV  P,  and  Y'  =  f  f  P. 

98.  CASE  VIH.  —  LET  THE  BEAM  BE  FIXED  AT  ONE  ENDJ  SUP- 

PORTED AT  THE  OTHER,  AND  UNIFORMLY  LOADED  OVER  ITS  WHOLE 
LENGTH. 


I       t 


FIG.  45.  FIG.  46. 

Take  the  origin  at  A,  Figs.  45  and  46,  and  the  notation  the 
same  as  in  the  preceding  cases,  then  equation   (50)  becomes 


El  -4  =  \wx -•  -  Vx (108) 

ii.    ! 

tu  *£.. 

fc 


114:  THE   RESISTANCE    OF   MATERIALS. 

Integrating  gives       =    L(x'-  ?)  +^J  P  -  «0,   (109) 

CS^-^);         (110) 


in  which  -^  =  0  for  x=  I.  and  w  =  0  for  a?  =  0. 
dx 

If  Y  =  0,  these  equations  become  the  same  as  those  under 
CASE  II. 

In  equation  (110)  y  is  also  zero,  for  x  =  I  ;  for  which  values 
we  have  Y  =  f  W=  f^Z  (111) 

This  value  substituted  in  equations  (108),  (109),  and  (110) 
gives  :  — 


(112) 


y  =      "  .(W-  3fo°  +  Px).         ......  (114) 

4oJiil 

The  point  of  maximum  deflection  is  found  by  placing  equa- 
tion (113)  equal  zero  and  solving  for  x.     This  gives 


and  this  in  (114)  gives 

y  =  A  =0.0054^j*.  (115) 

There  are  two  maxima  strains  ;  one  for  x  =  I;  the  other  for 
x  =  f  I.  The  former  in  (112)  gives 

sPaj  =  iw?  =  iW?,  (116) 

&nd  the  latter  gives 

sPa  =  -  TJh-  WZ  =  -  iV  ^  nearly. 

The  point  of  contra-flexure  is  found  from  equation  (112)  to 
be  at  x  =  %l,  at  which  point  the  longitudinal  strains  are  zero, 
and  there  is  only  transverse  shearing.  (See  Article  84.) 

If  the  beam  is  supported  by  three  props,  which  are  in  the 
same  horizontal,  Fig.  46,  then  each  part  is  subjected  to  the  same 
conditions  as  the  single  beam  in  Fig.  45.  Hence,  if  "W  is  the  load 


FLEXURE.  115 

on  half  the  beam,  each  of  the  end  props  will  sustain  Y  =  f  W, 
(Eq.  (Ill) ),  and  the  middle  prop  will  sustain  2W-  f  W  =  fW. 

Such  are  the  teachings  of  the  "  common  theory."  But  the 
mathematical  conditions  here  imposed  are  never  realized.  It  is 
impossible  to  maintain  the  props  exactly  in  the  same  horizontal. 
As  they  are  elastic  they  will  be  compressed,  and  as  the  central 
one  will  be  most  .compressed,  the  tendency  will  be  to  relieve  the 
strain  on  it  and  throw  a  greater  strain  upon  the  end  supports. 
If  the  supports  be  maintained  in  the  same  horizontal,  the  results 
above  deduced  will  be  practically  true  for  very  small  deflec- 
tions, but  will  be  somewhat  modified  as  the  strains  approach  the 
breaking  limit. 

99.  CASE  ix. — LET  THE  BEAM  BE  FIXED  AT  BOTH  ENDS  AND 

A  W-EIGHT  REST  UPON  IT  AT  ANY  POINT. 

To  simplify  the  case,  sup- 
pose that  the  weight  rests 
at  the  middle  of  the  length. 

Let  the  beam  be  extend- 
ed over  one  support  and  a 
wreight,  Pj  rest  at  C,  sufficient 
to  make  the  curve  horizontal 
over  the  support  A.  We 
have  Y  =  P,  -f  JP.  FIQ.  47. 

Let  AC  =  ql. 

Then  for  the  curve  AD  we  have, 

=  P,  glx  —  £EV  +  (C,  =  0). 

VA/tV 

To  find  P,  observe  that  -?  =  0  f  or  x  —  \l ; 

dx 


This  reduces  the  preceding  equations  to  the  following  :  — 

-4«)  (117) 


and  by  integrating  again,  we  find  :  — 


116 


THE   RESISTANCE   OF   MATERIALS. 


FT3 


-     -    -     (120) 

There  is  no  algebraic  maximum  of  the  moment  of  strain  as 
given  in  the  second  member  of  equation  (117),  but  inspection 
shows  that  within  the  limits  of  the  problem  the  moment  is 
greatest  for  x  =  0  or  x  —  \l.  These  in  (117)  give  the  same  value, 
with  contrary  signs  ;  hence  the  moment  of  greatest  strain  is 

xPx  =  ±iPZ  -    (121) 

The  moment  is  zero  for  x  =  JZ. 

1OO.  CASE  x.  —  LET  THE  BEAM  BE  FIXED  AT  BOTH  ENDS  AND 

A  LOAD  UNIFORMLY  DISTRIBUTED    OVER   ITS   WHOLE   LENGTH. 


FIG.  48. 

The  notation  being  the  same  as  before  used,  we  have 
Y  =  P,  +  $wl. 

Let  ql  =  AC. 

The  equation  of  moments  is 

-  Vx  +  Y^l  +  x) 
—  %wlx  -f  P^Z. 

(JfU 

Integrating,  and  observing  that  -/-  =  0  f  or  x  =  0  ;  also  y  =  0  f  or 

d>x  • 

x  =  0,  and  we  have 


Ely  =  ^wx4 

But  -jf-  —  0  for  x  =  I  ;  also  y  ~  0  for  x  =  I  : 
dx 


'*      1~ 


FLEXURE.  117 

which  substituted  in  the  previous  equations  give  :  — 


1    WZ3 

(125) 


Making  VT  =  0  we  find  for  the  points  of  contra-flexure 


j  0.78871 
~  \  0.2113? 


at  which  points  there  is  no  longitudinal  strain,  but  a  transverse 
shearing  strain.     (See  Article  84.) 

The  maximum  moments  are  for  x  =  0  and  x  =  %l. 
For  x  =  0,  the  second  member  of  Eq.  (122)  gives  -^Wl.  (126) 


Hence  the  greatest  strain  is  over  the  support,  at  which  point 
it  is  twice  as  great  as  at  the  middle.  If  W=  P,  we  see  that 
the  strain  over  the  support  is  f  as  great  in  this  case  as  in  the 
former. 


-X 


118  THE   RESISTANCE    OF   MATERIALS. 

1O1.      RESULTS  COLLECTED. 


a 

RELATIVE 

j>  IH 

MAX.  DEFLEC- 

[NO. OF 
THE 
CASE. 

CONDITION  OF 

THE  BEAM. 

HOW 
LOADED. 

GENERAL 
MOMENT  OF 

FLEXURE. 

MAXIMUM 
MOMENT  OF 

STRESS. 

s  2 

TION  OR  CO- 
EFFICIENT OF 

El 

I. 

LOAD  AT 

FREE  END. 

Px. 
Eg.  (53). 

PL 

24 

Eq.    (57). 

FIXED  AT 

ONE  END. 

II. 

UNIFORM 

LOAD. 

Eg.  (58). 

m 

12 

iw. 

Eq.  (61). 

IV. 

AT  THE 

MIDDLE. 

Eg.  (71). 

iPL 

6 

^\73). 

SUPPORTED 

ENDS. 

V. 

UNIFORM. 

2V(74).' 

w 

3 

skW. 
Eq.  (76). 

AT  0.634? 

For  AD 

VII. 

FIXED  AT 

FROM 
FIXED  END. 

Eg.  (91). 
ForDB 

|(2f3~3)P?. 

4+ 

p 

ONE  END 

Eg.  (100). 

Eg.  (94). 

Eq.  (104). 

AND  SUP- 

THE  OTHER. 

W 

VIII. 

UNIFORM. 

Eg.  (112). 

Eg.  (116). 

3 

.%  (115). 

IX. 

AT  THE 

iP<*-4*) 

iK 

P 

MIDDLE. 

Eg.  (117). 

Eg.  (121). 

192- 

Eq.  (120). 

FIXED  AT 

"" 

BOTH  ENDS. 

W 

W 

X. 

UNIFORM. 

Eg.  (122). 

Eq.  (126). 

2 

384 

Eq.  (125). 

1O2.  REMARKS. — It  will  be  seen  that  the  greatest  strains 
in  the  1st  and  2d  cases  are  as  2  to  1 ;  and  the  same  ratio  holds 
in  the  4th  and  5th  cases ;  but  in  the  9th  and  10th  the  ratio  is  as 
3  to  2.  This  is  peculiar,  and  further  remarks  are  made  upon 


FLEXUKE.  119 

it  in  Article  119.  The  maximum  strains  in  Cases  VII.  and 
VIII.  do  not  occur  at  the  points  of  maximum  deflection. 

Although  the  moment  in  the  1st  case  is  to  that  in  the  2d  as  2 
to  1,  yet  the  deflections  are  as  8  to  3 ;  and  in  the  4th  and  5th 
cases  the  deflections  are  as  8  to  5. 

A  comparison  of  Cases  IV.  and  IX.  shows  the  advantage  of 
fixing  the  ends  of  the  beam.  The  same  remark  applies  .to  Cases 
V.  and  X.  In  the  former  cases  the  strain  is  only  one-half  as 
great  when  the  beam  is  fixed  at  the  ends  as  when  it  is  supported, 
and  in  the  latter  two-thirds  as  great. 

Other  interesting  results  may  be  seen  by  examining  the  table. 


1O3.  MODIFICATION  OF  THE  FORMULAS  FOR  DEFLEC- 
TION.— It  will  be  observed  that  the  general  form  of  the  expres- 
sion for  the  maximum  deflection  of  rectangular  beams  is 

PZ3 


=  constant  x 


EM3 


Prof.  W.  A.  Norton,  of  New  Haven,  Ct.,  has  made  experi- 
ments to  test  the  correctness  of  this  expression.  (See  Van  JPibs- 
tranoPs  Eclectic  Engineering  Magazine,  vol.  3,  page  70.)  Ac- 
cording to  his  experiments,  for  beams  supported  at  their  ends 
and  loaded  at  the  middle,  the  expression  should  be 


= 


For  the  pine  sticks  which  he  used  he  found  the  mean  value 
of  C  to  be 

c  =  0.0000094.  j  .  ; 

A  consideration  of  transverse  shearing  stress,  in  combina- 
tion with  the  stretching  and  compressing  of  the  fibres,  leads  to 
an  expression  of  this  form.  For,  as  we  have  before  seen,  the 
strain  is  evenly  distributed  over  the  whole  transverse  section, 
and  hence  the  deflection  will  vary  inversely  as  the  area,  or  as 
Id  ;  it  is  also  uniform  over  its  whole  length,  and  equal  JP  (see 
Example  2,  Article  84)  ;  and  hence  the  amount  of  deflection 
will  vary  as  |P;  and  the  total  deflection  at  the  middle  will 


120  THE   RESISTANCE   OF   MATERIALS. 

evidently  vary  as  the  length ;  or,  in  this  case,  as  \l.  Hence, 
the  total  deflection  due  to  transverse  shearing  is,  C— f|-  = 

C  P7 

-  -y-r,  which  is  the  same  form  as  that  given  by  Professor  Norton. 

The  same  form  of  expression  is  al&o  reached,  in  a  more  circuitous 
way,  by  Weisbach,  in  his  Mechanics  of  Engineering,  4th  edi- 
tion, vol.  1,  page  522  of  the  recent  American  edition. 

In  Professor  Norton's  formula  \  C  is  the  reciprocal  of  the  co- 
efficient of  elasticity  to  transverse  shearing  of  white  pine  /  hence 
the  coefficient  is  425,531  pounds.  The  mean  value  of  E  in  the 
above  experiments  was  found  to  be  1,42T,965  Ibs. ;  and  hence, 
in  this  case,  the  reciprocal  of  J-C  is  a  little  more  than  ^  of  E. 
Weisbach,  in  the  reference  above  given,  says :  "  The  coefficient 
of  elasticity  for  transverse  shearing  is  generally  assumed  to  be 
equal  to  -^-E." 

If  the  load  is  uniformly  distributed  over  the  whole  length, 
the  shearing  stress  on  any  section,  distant  x  from  the  end,  is 
%  wl  —  wx.  (See  Example  1,  Article  84.)  Hence  the  deflection  for 
an  element  of  length  of  a  rectangular  beam  due  to  this  cause,  is 

(kwl  —  wx)dx 

G   7~7 1 

vcu 
and  for  a  distance  x  this  becomes  by  simple  integration, 

C—  (l-x)x 

^d  '  ~1%<L 

and  for  half  the  length,  make  x  =  ^-Z,  and  the  expression  be- 
comes 

i.  w?      »Wl 


from  which  we  see  that  the  same  load,  distributed  uniformly 
over  the  whole  length,  produces  half  as  much  deflection  due  to 
transverse  shearing  as  the  same  load  concentrated  at  the  middle. 
Equation  (76)  when  corrected  for  this  effect  becomes 


from  which  we  see  that  if  the  depth  be  constant  the  deflection 


FLEXURE.  121 

due  to  transverse  shearing  will  be  more  apparent  compared  with 

that  due  to  the  other  cause,  as  the  piece  is  shorter.     If  the  piece 

is  very  long,  the  effect  due  to  C  is  comparatively  small.     If 

4 

Q  —  -JE,  as  assumed  by  Weisbach,  the  deflection  becomes 

Wl   pBP 

A     OT717     7       X~7o-J-12 


If  I  =  d  the  quantity  in  [  ]  becomes  J-f-12 ;  or  the  effect  due 
to  C  is  4£  of  that  due  to  E. 

If  I  =  2Qd,  the  effect  due  to  C  is  ^  nearly  of  that  due  to  E. 


1O4.    ADDITIONAL  PROBLEMS  WHICH  ARE  PURPOSELY 
LEFT  UNSOLVED. 

1.  Suppose  that  a  beam  is  supported  at  its  extremities,  and 
lias  two  forces  at  any  point  between.     In  this  case  the  curve 
between  the  support  and  the  nearest  force  will  have  one  equa- 
tion ;  the  curve  between  the  forces  another  ;  and  the  remaining 
part  a  third. 

2.  In  the  preceding  case,  if  the  forces  are  equal  and  equidis- 
tant from  the  supports,  the  curve  between  the  forces  will  be  the 
arc  of  a  circle. 

3.  Suppose  that  the  beam  is  uniformly  loaded  and  rests  on 
four  supports. 

4.  Suppose  that  the  beam  is  supported  at  its  extremities  and 
has  a  load  uniformly  increasing  from  one  support  to  the  other. 

5.  Suppose  that  the  beam  is  uniformly  loaded  over  any  por- 
tion of  its  length. 

6.  Suppose  that  ft  has  forces  applied  at  various  points. 
These  problems  will  suggest  many  others. 

7.  Suppose  that  a  beam  is  supported  at  several  points,  and 
loaded  uniformly  over  its  whole  length. 

Let  W  =  the  weight  between  each  pair  of  supports, 

Y,,  Ya,  Y8,  &c.,  be  the  reactions  of  the  supports,  counting 

from  one  end, 
and  let  the  distances  between  the  supports  be  equal. 


122  THE   RESISTANCE    OF   MATERIALS. 

Then  we  have : — 


No.  of 
Sup- 
ports. 

2 

y, 

V2 

V3 

V4 

V5 

V6 

V7 

V8 

va 

v«i 

iW 

iW 

Frac 

tional 

parts 

of  W. 

3 

I 

¥ 

1 

4 

-i4o 

it 

tt 

A 

5 

H 

II 

If 

II 

ri 

6 

H 

ft 

ti 

n 

H 

H 

7 

A 

ttl 

HI 

w 

HI 

HI 

* 

8 

-Mr 

Hi 

Hi 

HI 

Ul 

ill 

HI 

tt 

9 

Hl 

iti 

1U 

IH 

IH 

HI 

3.14. 

388 

IH 

HI 

10 

m 

m 

IH 

m 

«» 

m 

IH 

IB 

m 

m 

If  the  beams  and  props  were  perfectly  rigid,  all  but  the  end 
ones  would  sustain  "W",  and  the  end  ones  each  J  "W". 

It  may  be  shown  that,  for  any  number  of  equidistant  props, 
the  inclination  at  the  end  may  be  found  from  the  equation 


which  for  10  props  becomes 


.      153 
=  ; 


and  the  maximum  deflection  for  any  number  of  props  is 


A  =  (24Y.-7W) 


384EI 


1(D5.    BEAMS  OF    VARIABLE    SECTIONS. 

For  these  I  is  variable,  and  its  value  must  be  substituted  in 


FLEXURE. 


123 


equation  (50)  before  the  integration  can  be  performed.     As  an 

example,  let   the    beam    be 

fixed  at  one  extremity,  and 

a  weight,  P,  be  suspended  at 

the  free  extremity,  Fig.  50. 

Let  the  breadth  be  constant, 

and  the  longitudinal  vertical 

sections  be  a  parabola.   Then 

all    the    transverse   sections 

will  be  rectangles. 

Let  I  =  the  length,  FlG- 50- 

1}  =  the  breadth,  and 
d  =  the  depth  at  the  fixed  extremity. 

If  y  is  the  whole  variable  depth  at  any  point,  we  have,  from 
the  equation  of  the  parabola, 

(-J?/)2  =j}x,  or  J^2  =  pi,  :.  p  =  27,  in  which  p  is  the  parameter 
of  the  parabola. 


(127) 


From  equation  (51)  we  have 

I  —  TV%3,  in  which  substitute  ?/,  from  equation  (127),  and  we 

=  Wy<*-- ,-    -    (128) 

The  equation  of  moments  is,  see  equation  (50), 

El  -A  —  IX  in  which  substitute  I,  from  equation  (128),  and 
we  have 


dx*  ~  Ebd* 


dy 


Multiply  by  the  dx  and  integrate,  observing  that  -,-  =  Of  or  a?  ==  I 
and  we  have 


dx       EM' 


124:  THE   RESISTANCE   OF   MATERIALS. 

Integrating  again  gives 

« 

y  = 


y  is  zero  for  x  —  0. 
y  =  A    for  x  =  I ; 
8P? 

•    •      ^-A      T^7~73    -          -          -          - 


If,  in  equation  (57),  we  substitute  I  = 
comes 


-     -     (129) 
(Eq.  (51)),  it  be- 


_ 

"  Ebd? 


which  is  one-half  that  of  (129)  ;  hence  the  deflection  of  a  prismatic 
beam  is  one-half  that  of  a  parabolic  beam  of  the  same  length, 
breadth,  and  greatest  depth,  when  fixed  at  one  end  and  free  at 
the  other,  and  has  the  same  weight  suspended  at  the  free  end. 

In  a  similar  manner  the  equation  of  the  curve  may  be  found 
for  any  other  form  of  beam,  if  the  law  of  increase  or  decrease 
of  section  is  known.  Several  examples  may  be  made  of  beams 
of  uniform  strength,  which  will  be  given  in  Chapter  VII. 

1O0.    BEAMS  SUBJECTED  TO    OBLIQUE  ^TRAINS.—  Let    the 

beam  be  prismatic,  fixed  at  one  end,  and 
support  a  weight,  P,  at  the  free  end;   the 
beam  being  so  inclined  that  the  direction  of 
the  force  shatt  make  an  obtuse  angle  with  the 
axis  of  the  beam,  as  in  Fig.  51. 
Let  PI  =  P  sin  9  =  component  of  P  per- 
pendicular to  the  axis  of  the  beam, 
and 
Pa  =  P  cos  9  —  component  parallel  to 

the  axis  of  the  beam. 
Take  the  origin  at  the  free  end,  the  axis 
of  x  being  parallel  to  the  axis  of  the  beam? 
and  y  perpendicular  to  it. 
Then  equation  (50)  becomes 


FIG.  61. 


T>  T> 

in  which  p5  =  -j  ;  and  q1  =  =?. 
dix  III.) 


(130) 


The  complete  integral  of  (130)  is  (see  Appen- 


FLEXURE. 


•125 


qx 


—qx       tf 

e        +  —J 


The  conditions  of  the  problem  give 

^7^, 

=  0  for  *  =  Z; 


. 
dx 

y  =  0  f  or  x  =  0  ;  and  these  combined  with  the  preceding  equation 


give  :- 


/ 

=  g  ( 


—qi 


0  =  d  +  C7 


3 


From  which  Ci  and  Ca  may  be  found,  and  the  equation  becomes  completely 
known. 
We  also  have  y  —  A  for  x  —  I- 

ql          —  ql         i^i 

.'.  A  -—  de     +  Crf    +  ~-l  ; 

Next,  suppose  that  the  force  makes  an  acute  angle  with  the  axis  of  the  beam,  as 
in  Fig.  52. 

For  the  sake  of  variety,  take  the  origin  at  A,  the  fixed  end,  #,  still  coin- 
ciding with  the  axis  of  the  beam  before  flexure.  Using  the  same  notation  as 
in  the  preceding  and  other  cases,  we  have 


(131) 


The  complete  integral  is 


FIG.  52. 


-(Z-z)    (132) 


in  which  A  and  B  are  arbitrary  constants. 
From  the  problem  we  have 
y  =  0  f  or  x  =  0 ; 

^•=0 for z  =  0;  and 

y  =  A  f  or  x  —  I ; 

by  means  of  which  the  equation  becomes 
completely  known. 


From  these  ^examples  we  see  how  easily  the  problem  is  com- 
plicated. One  difficulty  in  applying  these  cases  in  practice  is 
in  determining  the  value  of  I.  Before  it  can  be  determined, 
the  position  of  the  neutral  axis  must  be  known.  According  to 
Article  78,  3d  case,  it  appears  that  the  neutral  axis  does  not 
coincide  with  the  axis  of  the  beam.  Indeed,  according  to  the 
same  article,  it  is  not  parallel  to  the  axis,  and  hence  I  is  varia- 
ble, and  the  equations  above  are  only  a  secondary  approxima- 
tion ;  the  first  approximation  being  made  in  establishing  equa- 


120 


THE   RESISTANCE    OF   MATERIALS. 


tion  (50),  and  the  next  one  in  assuming  I  constant.  All  writ- 
ers, within  the  author's  acquaintance,  who  have  investigated  this 
and  similar  cases,  beginning  with  Navier,  have  assumed  that  I  is 
constant,  and  that  the  neutral  axis  coincides  with  the  axis  of 
the  beam.  These  assumptions  may  be  admissible  in  any  prac- 
tical case  were  extreme  accuracy  is  not  desired.  Many  other 
practical  examples  might  be  given,  the  solutions  of  which  are 
more  difficult  than  the  preceding ;  but  enough  have  been 
given  to  illustrate  the  methods. 


Y 


1O7,  FLEXURE  OF  COLUMNS. 

— If  a  weight  rests  upon  the  axis 
of  a  perfectly  symmetrical  and 
homogeneous  column,  we  see  no 
reason  why  it  should  bend  it ; 
but  in  practice  wre  know  that  it 
will  bend,  however  symmetrical 
and  homogeneous  it  may  be,  and 
however  carefully  the  weight  may 
be  placed  upon  it.  If  the  weight 
be  small,  the  deflection  may  not  be 
visible  to  the  unaided  eye.  If  the 

weight  is  not  so  heavy  as  to  crush  FlQ.  54.  FlQ.  53. 

the  column,  an  equilibrium  will  be  established  between  the 
weight  and  the  elastic  resistances  within  the  beam.  Let  the  col- 
umn rest  upon  a  horizontal  plane,  and  the  weight  P  on  the  upper 
end  be  vertically  over  the  lower  end.  Take  the  origin  of  coor- 
dinates at  the  lower  end  of  the  column,  Fig.  53,  a?  being  ver- 
tical, and  y  horizontal.  They  must  be  so  taken  here,  because  x 
was  assumed  to  coincide  with  the  axis  of  the  beam  when  equa- 
tion (50)  was  established.  Then  y  being  the  ordinate  to  any 
point  of  the  axis  of  the  column  after  flexure,  the  moment  of 
P  is  Py,  which  is  negative  in  reference  to  the  moment  of  re- 
sisting forces,  because  the  curve  is  concave  to  the  axis  of  x,  in 
which  case  the  ordinate  and  second  differential  coefficient  must 
have  contrary  signs  (Dif.  Cal.).  Hence  we  have, 


(133) 


FLEXURE.  127 

Multiply  by  dy  and  integrate  (observing  that  dx  is  constant), 
and  find 


But  -^  =  0  for  y  =  A  =  the  maximum  deflection.     These 
dx 

PA" 

values  in  the  preceding  equation  give  C(  =  -^p  which  being 
substituted  in  the  same  equation  and  reduced  gives 


But  y  —  0  for  x  =  0  /.  (73  =  0.     Hence  the  preceding  gives 

IT? 

y  =  A  sin  ^J-gj  ®     .......     (134) 

But  y  =  0  for  x  =  L     Therefore,  if  w  is  an  integer,  these  val- 
ues  reduce  (134)  to 


This  value  of  P  reduces  (134)  to 

x 
y  —  A  sin  rnr  j 

which  is  the  equation  of  the  curve.     It  is 
the  length  of  the   column  and  the   maximum   deflection. 
n  —  1,  the  curve  is  represented  by  a,  Fig.  54 ;    if  n  —  2,  by 
I ;  if  n  =  3,  by  c. 

If  n  —  1,  equation  (135)  becomes 

p  =  *  EI     .     _     .     ....    (136) 
which  is  the  formula  to  be  used  in  practice.     We  see  that  the 


128  THE   RESISTANCE   OF   MATERIALS. 

resistance  is  independent  of  the  deflection.     If  the  column  is 
cylindrical,  I  =  J  v  r*  (see  equation  (52)  )  ; 


(137) 


hence  the  resistance  varies  as  the  fourth  power  of  the  radius  (or 
diameter),  and  inversely  as  the  square  of  the  length.  If  the 
column  is  square,  I  =  T^  If  (equation  (51)  ), 


These  formulas,  according  to  Navier  *  and  Weisbach,f  should 
be  used  only  when  the  length  is  20  times  the  diameter  for 
cylindrical  columns,  or  20  times  the  least  thickness  for  rectangular 
columns ;  and  Navier  says  that  for  safety  only  ^  of  the  calcu- 
lated weight  should  be  used  in  case  of  wood,  and  J  to  -§-  in  case  of 
iron ;  but  Weisbach  says  they  should  have  a  twenty-fold  security. 

Examples  1. — What  must  be  the  diameter  of  a  cast-iron  column,  whose  length 
is  12  feet,  to  sustain  a  weight  of  30  tons  (of  2,000  Ibs.  each) ;  E  =  16,000,000 
Ibs. ;  and  factor  of  safety  •£$.  Ans.  d  =  7.52  in. 

2.  If  the  column  be  square  and  the  data  the  same  as  in  the  preceding  exam- 
ple, equation  (138)  gives 

_^4/  12  X  60,000  X  (12  X  12)*  X  20  _ 


In  the  analysis  of  this  problem  I  have  followed  the  method  of 
Javier ;  but  as  it  is  well  known  that  the  results  are  not  relied  upon 
by  practical  men,  I  have  given  only  one  case.  For  other  cases 
see  Appendix  III.  There  are  some  reasons  for  the  failure  of 
the  theory  which  are  quite  evident,  but  it  is  not  easy  to  remedy 
them  ;  and  for  this  reason  the  empyrical  formulas  of  Article  52 
are  much  more  satisfactory.  It  will  be  observed  that  the  law  of 
strength,  as  given  in  the  formulas  in  that  article,  are  the  same  as 
those  given  in  equations  (137)  and  (138)  for  -wooden  columns, 
and  nearly  the  same  as  for  iron  ones.  The  chief  difference  is  in 
the  coefficients,  or  constant  factors.  In  the  analysis  it  was  as- 
sumed that  the  neutral  axis  coincides  with  the  axis  of  the  beam, 
but  it  is  possible  for  the  whole  column  to  be  compressed,  although 
much  more  on  the  concave  than  on  the  convex  side,  in  which 


*  Navier,  Resume  des  Legons,  1839,  p.  204. 

f  Weisbach's  Mechanics  and  Engineering.     Vol.  1,  p.  219.     1st  Am.  ed. 


FLEXUEE.  129 

case  the  neutral  axis  would  be  ideal,  having  its  position  entirely 
outside  the  beam  on  the  convex  side.  In  this  case,  if  the  ideal 
axis  is  parallel  to  the  axis  of  the  beam,  it  does  not  affect  the  form 
of  equation  (136),  but  it  does  affect  the  value  of  I ;  and  hence 
the  values  of  equations  (137)  and  (138).  The  problem  of  the 
flexure  of  columns  is  then  more  interesting  as  an  analytical  one 
than  profitable  as  a  practical  one. 

GRAPHICAL  METHOD. 

1O8.  THE  GRAPHICAL,  METHOD  consists  in  representing 
quantities  by  geometrical  magnitudes,  and  reasoning  upon  them, 
with  or  without  the  aid  of  algebraic  symbols.  This  method  has 
some  advantages  over  purely  analytical  processes;  for  by  it 
many  problems  which  involve  the  spirit  of  the  Differential  and 
Integral  Calculus  may  be  solved  without  a  knowledge  of  the 
processes  used  in  those  branches  of  mathematics  ;  and  in  some 
of  the  more  elementary  problems,  in  which  the  spirit  of  the 
Calculus  is  not  involved,  the  quantities  may  be  directly  presented 
to  the  eye,  and  hence  the  solutions  may  be  more  easily  retained. 
It  is  distinguished,  in  this  connection,  from  pure  geometry  by 
being  applied  to  problems  which  involve  mechanical  principles, 
and  to  use  it  profitably  in  such  cases  requires  a  knowledge  of 
the  elementary  principles  of  mechanics  as  well  as  of  geometry. 

But  graphical  methods  are  generally  special,  and  often  re- 
quire peculiar  treatment  and  much  skill  in  their  management. 
It  is  not  so  powerful  a  mode  of  analysis  as  the  analytical  one, 
and  those  who  have  sufficient  knowledge  of  mathematics  to  use 
the  latter  will  rarely  resort  to  the  former,  unless  it  be  to  illus- 
trate a  principle  or  demonstrate  a  problem  for  those  who  cannot 
use  the  higher  mathematics.  A  few  examples  will  now  be  given 
to  illustrate  the  method. 


1O9.  GENERAL.  PROBLEM  OF  THE   DEFLECTION  OP 

BEAMS. — To  find  the  total  deflection  of  a  prismatic  beam  which 
is  bent  ~by  a  force  acting  normal  to  the  axis  of  the  beam 
without  the  aid  of  the  Calculus. 

Let  a  beam,  AB,  Fig.  55,  be  bent  by  a  force,  P,  in  which 
case   the  fibres  on  the   convex   side  will  be  elongated,  and 
those  on  the  concave  side  will  be  compressed.     Let  AB  be  the 
9 


130 


THE    RESISTANCE    OF    MATEEIALS. 


FIG.  55. 


ON: 


::  LN:  Jce  =  *  = 


neutral  axis.      Take  two  sections  normal  to  the  neutral  axis 

at  L  and  N,  which  are  in- 
definitely near  each  other. 
These,  if  prolonged,  will 
meet  at  some  point  as  O. 
Draw  KN  parallel  to  LO. 
Then  will  Jce,  =  A,  be  the 
distance  between  KN  and 
EN  at  Jc,  and  is  the  elonga- 
tion of  the  fibre  at  Jc.  Let 
eN  =  y,  then  from  the 
similar  triangles  tiNe  and 
LON  we  have 

kLN        LN 
ON        ~~  ON  y' 
If,  now,  we  conceive  that  a  force  p,  acting  in  the  direction  of 
the  fibres,  or,  which  is  the  same  thing,  acting  parallel  to  the 
axis  of  the  beam,  is  applied  at  Jc  to  elongate  a  single  fibre,  we 
have,  from  equation  (3)  and  the  preceding  one, 

™  _  Ice         E  ^ 
p  =  jii 


in  which 
tion  turns 
force  is 


which  is  found  by  multiplying  the  force  by  the  perpendicu- 
lar y. 

This  is  the  moment  of  a  force  which  is  sufficient  to  elongate 
or  compress  any  fibre  whose  original  length  was  LN",  an  amount 
equal  to  the  distance  between  the  planes  KN  and  EN  measured 
on  the  fibre  or  fibre  prolonged.  Hence,  the  sum  of  all  the 
moments  of  the  resisting  forces  is 

E 


LN 

%  is  the  transverse  section  of  the  fibre.     As  the  see- 
about  N  on  the  neutral  axis,  the  moment  of  this 


E       ,  r- 


in  which  2  denotes  summation  ;  and  in  the  first  member  means 
that  the  sum  of  the  moments  of  all  the  forces  which  elongate 
and  compress  the  fibres  is  to  be  taken  ;  and  in  the  second  mem- 


FLEXURE.  131 


ber  it  means  that  the  sum  of  all  the  quantities  y1  A«  included  in 
the  transverse  section  is  to  be  taken.  The  quantity,  Sy2  &a  is 
called  the  moment  of  inertia,  which  call  I. 

But  the  sum  of  the  moments  of  the  resisting  forces  equals 
the  sum  of  the  moments  of  the  applied  forces.  Calling  the  latter 
sPX,  in  which  X  is  the  arm  of  the  force  P,  and  we  have 


In  the  figure  draw  U  tangent  to  the  neutral  axis  at  L,  and 
N«  tangent  at  N.  The  distance  ab,  intercepted  by  those  tan- 
gents on  the  vertical  through  A,  is  the  deflection  at  A  due  to  the 
curvature  between  L  and  N".  As  LN  is  indefinitely  short,  it 
may  be  considered  a  straight  line,  and  equal  x  ;  and  U  =  LC 
very  nearly  for  small  deflections  ;  and  LC  =  X.  (L  stands  for 
two  points.) 

By  the  triangles  OLK  and  a'Lb,  considered  similar,  we  have 


in  which  substitute  ON  from  equation  (13,9)  and  we  have 
,      Xx  sPX 


which  is  sufficiently  exact  for  small  deflections.  If,  now,  tan- 
gents be  drawn  at  every  point  of  the  curve  AB,  they  will  divide 
the  line  AC  into  an  infinite  number  of  small  parts,  the  sum 
of  which  will  equaLthe  line  AC,  the  total  deflection.  But  the 
expression  for  the  value  of  each  of  these  small  spaces  will  be  of 
the  same  form  as  that  given  above  for  ab,  in  which  P,  E,  and  I 
are  constant. 

This  is  as  far  as  we  can  proceed  with  the  general  solution. 
AVe  will  now  consider 

PARTICULAR  CASES. 

I1O.     CASE    1.-L.ET   THE    BEAM   BE    FIXED    AT  ONE  END, 
AND  A  LOAD,  P,  BE  APPLIED  AT  THE  FREE  END.  -  Tills  IS    a 

part  of  Case  I.,  page  99,  and  Fig.  37  is  applicable.  The  moment 


132  THE   RESISTANCE    OF   MATEEIAL6. 

of  P,  in  reference  to  any  point  on  the  axis,  is  PX.    Hence  sPX 
is  simply  PX,  which,  substituted  in  equation  (140),  gives 

p 

ab  =  ,^-f  X2a? 
Jii.l 


,-^sX2^    -    -  (141) 

This  equation  has  been  deduced  directly  from  the  figure.  It 
now  remains  to  find  the  sum  of  all  the  values  of  X2a?,  which 
result  from  giving  to  X  all  possible  values  from  X  =  0  to 
X  =  I*.  To  do  this,  construct  a  figure  some  property  of  which 
represents  the  expression,  but  which  has  not  necessarily  any 
relation  to  the  problem  which  is  being  solved.  If  X  be  used 
as  a  linear  quantity,  X2  may  be  an  area  and  X2#  will  be  a  small 
volume.  These  conditions  are  represented  by  a  pyramid,  Fig. 
56,  in  which 

AB  —  I—  the  altitude,  and  the  base  BCDE  is  a  square,  whose 
sides,  BC  and  CD,  each  =  I.     Let  bcde  be 
a  section  parallel  to  the  base,  and  make 
'another  section  infinitely  near  it,  and  call 
the  distance  between  the  two  sections  x. 

Then  A&  —  ^L=.~bc  =  cd, 

X2  —  area  l>cde,  and 
X2a?  =  the  volume  of  the  la- 
mina bede, 

which  is  the  expression  sought.     The  sum 
of  all  the  laminae  of  the  pyramid  which  FlG-  56- 

are  parallel  to  the  base  is  limited  by  the  volume  of  the  pyramid, 
and  this  equals  the  value  of  the  expression  sX2#  between  the 
limits  0  and  I.  The  volume  of  the  pyramid  is  the  area  of  the 
base  (=  Z2)  multiplied  by  one-third  the  altitude  (|£),  or  ^Z3, 
which  is  the  value  sought. 

Hence,  AC  =  gg-j 

wfech  is  the  same  as  equation  (57). 

The  value  of  X2a?  may  also  be  found  by  statical  moments  as 
follows: — Let  ABC,  Fig.  57,  be  a  triangle,  whose  thick- 

*  This  by  the  calculus  becomes  /    x'2  dx  —  \P 

J  o 


FLEXUKE.  133 

ness   is   unity,   and  which  is  acted  upon  by  gravity  (or  any 
other  system  of  parallel  forces  which  is  the  same 
on  each  unit  of   the  body).     Take  an   infinitely 
thin  strip,  be,  perpendicular  to  the  base,  and 
let  AB  =  I  =  BC, 


p  —  the  weight  of  a  unit  of  volume. 
Then  Xcc  =  the  area  of  the  infinitely  thin  strip  be,  and 
—  the  weight  of  the  strip  be,  and 
=  the  moment  of  the  strip,  when  A  is  taken  as  the 
origin  of  moments.   If  the  weight  of  a  unit  of  volume  be  taken 
as  a  unit,  the  moment  becomes  X2#,  which  is  the  quantity  sought, 
and  the  value  of  sX2#  from  0  to  /is  the  moment  of  the  whole 
triangle  ABC.     Its  area  is  %l\  and  its  centre  of  gravity  f  I  to  the 
right  of  A.     Hence  the  moment  is  %l3  as  before  found.* 

Ill,    CASE  II.  —  LET  THE  BEAM  BE  FIXED    AT    ONE  END, 
AND    UNIFORMLY    LOADED    OVER    ITS    WHOLE     LENGTH.  - 

This  is  the  same  as  a  part  of  Case  II.,  page  101,  and  Fig.  39  is 
applicable. 

Let  X  be  measured  from  the  free  end,  and 
w  =  the  load  on  a  unit  of  length  ;  then 
w~K  —  the  load  on  a  length  X,  and 
-iX  =  the  distance  of  the  centre  of  gravity  of  the  load  from 

the  section  which  is  considered. 

Hence  the  moment  is  |wX2,  which  equals  sPX,  and  equation 
(140)  becomes 

ab  =          X3#  and 


w   x~l  r, 
AC  —  T^F^T  -2X8a?  —  the  total  deflection. 


To  find  the  value  of  2X3#,  observe,  in  Fig  56,  that  X'o;  is 
the  volume  of  the  lamina  bede,  and  this  multiplied  by  the  alti- 
tude of  A.—  bcde,  wrhich  is  X,  gives  X3#,  the  expression  sought. 
Hence  the  sum  sought  is  the  volume  of  the  pyramid  A—  BCDE, 

x  =  I 

*  This  may  be  written  2  X2z  =  JaZs. 


134  THE   RESISTANCE    OF   MATERIALS. 

multiplied  by  the  distance  of  the  centre  of  gravity  of  the  pyra- 
mid from  the  apex ;  or, 


where  W,  is  the  total  load  on  the  beam. 

1 12J.  CASE  III. — LET  THE  BEAM  BE  SUPPORTED  AT  ITS 
ENDS  AND  LOADED  AT  THE  MIDDLE  BY  A  WEIGHT  P,  as  in 

Fig.  40.     The  reaction  of  each  support  is  ^P,  and  the  moment 
is  !-PX,  and  equation  (140)  becomes 

P 

But  in  this  case  the  greatest  deflection  is  at  the  middle,  and 
the  limits  of  sX2a?  are  0  and  \l.  Hence,  in  Fig.  56,  let  the 
altitude  of  the  pyramid  be  JZ,  and  each  side  of  the  base  also  JZ, 
and  the  volume  will  be 


48EJ' 
which  is  the  same  as  equation  (73). 

113.  CASE  IV.—  LET  THE  BEAM  BE  SUPPORTED  AT  ITS 
ENDS  AND  UNIFORMLY  LOADED,  AS  IN  FIG.  42. 

w  being  the  load  on  a  unit  of  length,  the  reaction  of  each 
support  is  ^wl,  and  its  moment  at  any  point  of  the  beam  is 
%wlK.  On  the  length  X  there  is  a  load  wX,  the  centre  of 
which  is  at  J-X  from  the  point  considered  ;  hence  its  moment  is 
£wX2,  and  the  total  moment  is  the  difference  of  these  moments  ; 

.-.  sPX  =  tyolX.  - 
and  equation  (140)  becomes 


and  the  total  deflection  at  the  middle  is, 

«/, 

AC  =± 


x=Q 


FLEXURE.  135 

The  values  of  the  terms  within  the  parentheses  have  already 
been  found,  and  by  subtracting  them  we  have 


AP- 

-384EJ 

114.    HEM  ARK     ABOUT     OTHER     CASES.  -  This     method, 

which  appears  so  simple  in  these  cases,  unfortunately  becomes 
very  complex  in  many  other  cases,  and  in  some  it  is  quite  pow- 
erless. To  solve  the  9th  and  10th  cases,  pages  115  and  116,  ne- 
cessitates an  expression  for  the  inclination  of  the  curve,  so  that 
the  condition  of  its  being  horizontal  over  the  support  may  be 
imposed  upon  the  analysis.  But  the  9th  case  may  be  easily 
solved  if  we  find  by  any  process  that  the  weight  which  must 
be  suspended  at  the  outer  end  of  the  beam  to  make  it  horizontal 
over  the  support  is  £P£  divided  by  AC,*  Fig.  47.  For,  the 
reaction  of  the  support  is  £P  +  P  ; 

.-.  sPX  -  P/AC  +  X)  - 


••  EJ          > 

p 

and  the  deflection  at  the  centre  =  •$•  ^-j  (fe~K.x  —  4sX2#)  taken 

between  the  limits  0  and  \l. 

The  part  sXa?  is  ^the  area  of  a  triangle  whose  base  and  alti- 
tude are  each  JZ,  /.  sX#  =  -JF,  and  sX2#  between  the  limits  0 

P    P 

and  ^,  is  ^  -/.  ACf  =  ' 


All  these  expressions  contain  I,  the  value  of  which  remains  to 
be  found  by  the  graphical  method. 


*  This  "  AC  "  refers  to  Fig.  47. 
f  This  "  AC  "  refers  to  Fig.  55.  , 


136 


THE   RESISTANCE    OF   MATERIALS. 


115.    MOMENT  OF  INERTIA  OF  A  RECTANGLE.  — 

the  moment  of  inertia  of  a  rectangle  about  one  end  as  an  axis. 

Let  ABCD,  Fig.  58,  be  a  rectangle.     Make  BG  perpendicu- 
lar to  and  equal  AB,  and  complete  the 
wedge  G  -  ABCD. 

Let  A0  =  the  area  of  a  very  small  sur- 

face at  E,  and 
y  =  AE  =  EF,  then 
2/A#  =  the  volume  of  a  very  small 
prism  EF,  and  this  multi- 
plied by  y  gives 

ya  A#  =  the  moment  of  inertia  of 
the  elementary  area  at  E, 
which  is  also  the  statical 
moment  of  the  prism  EF,  and 
23/a  Aa  —  I  =  the  moment  of  inertia   of  the   rectangle 

ABCD. 

Hence  the  moment  of  inertia  of  the  rectangle  is  represented 
by  the  statical  moment  of  the  wedge  G  —  ABCD.     If 

",  and 


then  the  volume  of  the  wedge  is 

Id  x  \d  = 
and  the  moment  =  %W  x  %d  =  $cF     .....     (143) 

If  the  axis  of  moments  passes  through  the  centre  of  the  rec- 
tangle, and  parallel  to  one  end,  we  have  BE  =  GB  =  \d  in 
Fig.  59.  Hence  the  moment  of  inertia  of  the  rectangle  — 

2  x  b  x  \d  x  \d  x  |  of  \d  =  TV  &? 
which  is  the  same  as  equation  (50). 

116.    THE  MOMENT    OF  INERTIA  OF  A  TRIANGLE  about 

an  axis  parallel  to  the  base  and  passing  through  the  vertex  is, 
in  a  similar  way,  the  statical  moment  of  the  pyramid  ABCDE. 
Fig.  60. 

Let  5  —  CB  =  base  of  the  triangle,  and 

d  =  AB  =  BD  =  CE  ==  altitude   of   the  triangle  and 
pyramid  and  sides  of  the  base  of  the  pyramid. 
The  volume  of  the  pyramid  —  ^  fid*. 

The  centre  of  gravity  is  $d  from  the  apex,  consequently  the 
statical  moment  is  -J-  bd?  x  f  d  = 


FLEXUKE. 


137 


But  in  a  triangular  beam  the  neutral  axis  passes  through  the 
centre  of  gravity  of  the  triangle,  and  it  is  desirable  to  find  the 
moment  of  inertia  about  an  axis  which  passes  through  the  centre 
and  parallel  to  the  base. 

This  may  be  done  as  in  the  preceding  Article ;  but  it  may  be 


FIG.  59. 


FIG.  60. 


more  easily  done  by  using  fas  formula,  of  reduction,  which  is 
as  follows: — The  moment  of  inertia  of  a  figure  about  an  axis 
passing  through  its  centre  equals  the  moment  of  inertia  about  an 
axis  parallel  to  it,  minus  the  area  of  the  figure  multiplied  by 
the  square  of  the  distance  between  the  axes.  (See  Appendix  III.) 
This  gives  for  the  moment  of  inertia  of  a  triangle  about  an 
axis  passing  through  its  centre  and  parallel  to  the  base 

i^3  -  &d  X  (|^)3  =  -&!>(?  -     -     (143) 

117.    THE    MOMENT    OF    INERTIA    OF    A    CIRCLE    maybe 

represented  in  the  same  way,  but  it  is  not  easy  to  find  the  vol- 
ume of  the  wedge,  or  the  position  of  its  centre  of  gravity, 
except  by  analysis  which  is  more  tedious  than  that  required  to 
find  the  moment  directly,  as  was  done  in  equation  (51).  But  it 
may  be  found  practically,  by  those  who  can  only  perform 
multiplication,  as  follows : — Make  a  wedge-shaped  piece  out  of 
wood,  or  plaster  of  Paris,  or  other  convenient  material,  the  base 
of  which  is  the  semicircle  required,  and  the  altitude  is  the 
radius  of  the  circle ;  then  find  its  volume  by  im- 
mersing it  in  a  liquid  and  measuring  the  amount  of 
water  displaced.  Then  determine  the  distance  of 
the  centre  of  gravity  of  the  wedge  from  the  centre 
of  the  circle  by  balancing  it  on  a  knife  edge,  holding  a 

the  edge  of  the  knife  under  the  base  of  the  wedge,      FIG.  eo  a. 
and  parallel   to  the  edge,  ab,  of  the  wedge,  keeping  the  side 


138 


THE   RESISTANCE    OF   MATERIALS. 


vertical,  and  measuring  the  distance  between  the  edge  db  and 
the  line  of  support.  Then  the  statical  moment  is  the  product 
of  the  volume  multiplied  by  the  horizontal  distance  of  the 
centre  from  the  edge.  Its  value  for  the  whole  circle,  or  for 
both  wedges,  is  %*r*. 

There  are,  however,  many  methods  of  calculating  the  moment 
of  inertia  of  a  circle  without  using  the  Calculus.  The  following 
method,  which  the  author  has  devised,  appears  as  simple  as 
any  of  the  known  methods  :  — 

The  moment  of  inertia  of  a  circle  is  the 
same  about  all  its  diameters.  Hence  the 
moment  about  X  in  the  figure,  plus  the 
moment  about  Y,  equals  twice  the  moment 
about  X.  The  distance  to  any  point  A  is 
P,  and  equals  -v/#2  4-  y2  ;  or  f  —  x*  +  y*  ; 
and  if  A#  be  an  elementary  area,  as  be- 
fore,  we  have  Fie606 


the  latter  of  which  is  called  the  polar  moment  of  inertia,  in 
reference  to  an  axis  perpendicular  to  the  plane  of  the  circle,  and 
passing  through  its  centre  C.  To  find  the  value  of  2  Aa  pa, 
take  a  triangle  whose  base  and  altitude  are  each  equal  to  r,  the 
radius  of  the  circle,  and  revolve  it  about  the  axis  through  C, 
and  construct  an  infinitely  small  prism  on  the  element  Aa  as  a 
base. 
We  have  f  =  CA  =  AB,  Fig.  60  c. 

Aa?  =  volume  of  the  small  prism  AB. 

A#fCA=  A#/»2=the  moment  of  AB, 

the  form  of  the  quantity  sought. 
Hence  2  A#  f  is  the  product  of  the  volume  of  the 
solid  generated  by  the  triangle,  multiplied  by  the 
abscissa  of  its  centre  of  gravity  from  C.     The  solid  is  what  re- 
mains of  a  cylinder  after  a  cone  has  been  taken  out  of  it,  the 
base  of  the  cone  being  the  upper  base  of  the  cylinder,  and 
the  apex  of  which  is  at  the  centre  of  the  base  of  the  cylinder. 
Hence  the  volume  of  the  solid  is  the  volume  of  the  cylinder, 
less  the  volume  of  the  cone  ;  or  ^f  x  r  —  *r*  x  '  r  =  ^r*. 


FIG.  60  c. 


FLEXTTRE.  139 

If  now  the  solid  be  divided  into  an  infinite  number  of  pieces, 
by  planes  which  pass  through  its  axis,  each  small  solid  will  be  a 
pyramid,  having  its  vertex  at  C,  and  the  abscissa  to  the  centre 
of  gravity  of  each  is  \r  from  C.  Hence  we  finally  have 

2  A#  ?  =  f  *r*  x  f  r  —  £  ir4,  which 
equals  2  2  A#  #2. 

.:SMtf  =  krr\  (144) 

118.    MOMENT  OF  INERTIA  OF  OTHER  SURFACES. The 

general  method  indicated  in  the  preceding  articles  is  applicable 
to  surfaces  of  any  character,  and  with  careful  manipulation  ap- 
proximations may  be  made  which  will  be  very  nearly  correct, 
and,  as  we  have  seen  above,  in  some  cases  exact  formulas  may 
be  found. 


140 


THE   RESISTANCE    OF   MATERIALS. 


CHAPTER  YL 


TRANSVERSE  STRENGTH. 


119.      STRENGTH    OF    RECTANGULAR  BEAMS. The  the- 

ories  which  have  been  advanced  from  time  to  time  to  explain 
the  mechanical  action  of  the  fibres,  have  been  already  given 
in  Chapter  IY.  Both  the  common  theory,  and  Barlow's  theory 
of  "  the  resistance  to  flexure  "  will  be  considered  in  this  chap- 
ter. 

First,  consider  the  common  theory,  according  to  which  the 
neutral  axis  passes  through  the  centre  of  gravity  of  the  trans- 
verse sections,  and  the  strain  upon  the  fibres  is  directly  propor- 
tional to  their  distance  from  the  neutral  axis. 

Continuing  the  use  of  the  geometrical  method,  let  Fig.  61 
represent  a  rectangular  beam 
which  is  strained  by  a  force  P 
applied  at  any  point.  Let  de  be 
on  the  neutral  axis,  and  ab  repre- 
sent the  strain  upon  the  lowest 
fibre.  Pass  a  plane,  de-cb,  and 
the  wedge  so  cut  off  represents 
the  strains  on  the  lower  side,  and 
the  similar  wedge  on  the  other 
side  represents  the  strains  on  the 
upper  side. 

Let  R  —  the  strain  upon  a  FIG.  ei. 

unit  of  fibres  most  remote  from  the  neutral  axis  on  the  side 
which  first  ruptures,  on  the  hypothesis  that  all  the  fibres  of  the 
unit  are  equally  strained,  and  b  =  the  breadth  and  d  —  the 
depth  of  the  beam. 

Let  db  =  E  ;  then,  the  total  resistance  to  compression  =  %Rb 
x  \d  —  $Rbd,  =  the  volume  of  the  lower  wedge  ;  and  the  mo- 


TRANSVERSE  STRENGTH.  141 

ment  of  resistance  is  this  value  multiplied  by  the  distance  of 
the  centre  of  gravity  of  the  wedge  from  de,  which  is  f  of  \d  = 
\d  j  consequently  the  moment  is 


and  as  the  moment  of  resistance  to  tension  is  the  same,  the  total 
moment  of  resistance  is  —  . 

£&?,•==:  SLfcy  (145) 

which  equals  the  moment  of  the  applied  or  bending  forces. 

If  the  beam  be  fixed  at  one  end  and  loaded  by  a  weight,  P,  at 
the  free  end,  we  have  for  the  dangerous  section,  or  that  most 
liable  to  break, 

PI  =  fR&f  . 

In  rectangular  beams  the  dangerous  section  will  be  where  the 
sum  of  the  moments  of  stresses  is  greatest,  the  maximum  values 
of  which  for  a  few  cases  are  given  in  a  table  on  page  118.  Using 
those  values,  and  placing  them  equal  to  ^RbcF,  and  we  have  for 
solid  rectangular  beams  at  the  dangerous  section,  the  following 
formulas  :  — 

FOR  A  BEAM  FIXED  AT  ONE  END  AND  A  LOAD,  P,  AT  THE  FREE 

END  ;  PI  =  fKbd?  ;    ......  (146) 

AND  FOR  AN  UNIFORM  LOAD  ;   %Wl  —  -j-Rfo^  -     -     (147) 

FOR  A  BEAM  SUPPORTED  AT  ITS  ENDS  AND  A  LOAD,  P,  AT  THE 

MIDDLE  ;  %Pl  =  $RkF  ;      -    -  ...     (148) 

AND  FOR  AN  UNIFORM  LOAD  ;  £W7.  =  ^RbcP  •       -     (149) 

AND    FOR  A  LOAD  AT  THE  MIDDLE,  AND  ALSO  AN  UNIFORM  LOAD  ; 

i(2P  +  WX  =  |RfoF         .........     (150) 

FOR   A   BEAM   FIXED    AT  BOTH    ENDS    AND   A    LOAD,    P,    AT    THE 

MIDDLE  ;  JPZ  =  fRta";      -  (151) 

AND  FOR  AN  UNIFORM  LOAD,  END  SECTION  ;  -fyWl  —  fS&xF  ;    (152) 

MIDDLE  SECTION  ;  -faWl  =  fRfof       ....          (153) 

These  expressions  show  that  in  solid  rectangular  beams  the 
strength  varies  as  the  breadth  and  square  of  the  depth,  and 
hence  breadth  should  be  sacrificed  for  depth.  In  all  the  cases', 
except  for  a  beam  fixed  at  the  ends,  it  appears  that  a  beam  will 
support  twice  as  much  if  the  load  be  uniformly  distributed  over 
the  whole  length  as  if  it  be  concentrated  at  the  middle  of  the 
length.  The  case  in  which  a  beam  is  fixed  at  both  ends  and 
loaded  at  the  middle  has  given  rise  to  considerable  discussion, 


142  THE   RESISTANCE    OF   MATERIALS. 

for  it  is  found  by  experiment  that  a  beam  whose  ends  are  fixed 
in  walls  or  masonry  will  not  sustain  as  much  as  is  indicated  by 
the  formula,  and  also  that  it  requires  considerably  more  load 
to  break  it  at  the  ends  than  at  the  middle,  but  the  analysis 
shows  that  it  is  equally  liable  to  break  at  the  ends  or  at  the 
middle.  But  it  should  be  observed  that  there  is  considerable 
difference  between  the  condition  of  mathematical  fixedness,  in 
which  case  the  beam  is  horizontal  over  the  supports,  and  that  of 
imbedding  a  beam  in  a  wall.  For  in  the  latter  case  the  deflec- 
tion will  extend  some  distance  into  the  wall. 

Mr.   Barlow  concludes  from   his  experiments  that  equation 
(151)  should  be 

|PZ  =  £R&F     -    -    -    -  -     (15.4) 

and  this  relation  is  doubtless  more  nearly  realized  in  practice 
than  the  ideal  one  given  above.  In  either  case,  it  appears  that 
writers  and  experimenters  have  entirely  overlooked  the  effect 
due  to  the  change  of  position  of  the  neutral  axis,  which  must 
take  place.  It  has  been  assumed  that  the  neutral  axis  coincides 
with  the  axis  of  the  beam,  and  that  its  length  remains  unchanged 
during  flexure  ;  but  if  the  ends  of  the  beam  are  fixed,  the  axis 
must  be  elongated  by  flexure,  or  else  approach  much  nearer  the 
concave  than  the  convex  side,  or  both  take  place  at  the  same 
time,  in  which  case  the  moment  of  resistance  will  not  be  ^Rld*. 
The  phenomena  are  of  too  complex  a  character  to  admit  of  a 
thorough  and  exact  analysis,  and  it  is  probably  safer  to  accept 
the  results  of  Mr.  Barlow  in  practice  than  depend  upon  theoreti- 
cal results. 


MODULUS  OF  RUPTURE.  —  When  a  beam  is  support- 
ed at  its  ends,  and  loaded  uniformly  over  its  whole  length,  and 
also  loaded  at  the  middle,  we  find  from  equation  (150) 


E 

~ 


in  which  W  may  be  the  weight  of  the  beam.  Beams  of  known 
dimensions,  thus  supported,  have  been  broken  by  weights  placed 
at  the  middle  of  the  length,  and  the  corresponding  value  of  R 
has  been  found  for  various  materials,  the  results  of  which  have 
been  entered  in  the  table  in  Appendix  IY.  This  is  called  the 
MODULUS  OF  RUPTURE,  and  is  defined  to  be  the  strain  upon  a 


TRANSVERSE  STRENGTH.  143 

square  inch  of  fibres  most  remote  from  the  neutral  axis  on  the 
side  which  first  ruptures.  It  would  seem  from  this  definition 
that  K  should  equal  either  the  tenacity  or  crushing  resistance  of 
the  material,  depending  upon  whether  it  broke  by  crushing  or 
tearing,  but  an  examination  of  the  table  shows  the  paradoxical 
result  that  it  never  equals  either,  but  is  always  greater  than  the 
smaller  and  less  than  the  greater.  For  instance,  in  the  case  of 
cast  iron :  — 

The  mean  value  of  T  =  16,000  Ibs. 
«  "  C  =  96,000   " 

"  «  K  —  36,000   "     nearly, 

hence  K  is  about  2J  times  T,  and  a  little  over  -}  of  C. 
For  English  oak — 

T  =  17,000  Ibs. ; 

C  =    9,500  Ibs. ;  and 

E  =  10,000  Ibs. ; 

hence  E  exceeds  C,  and  is  more  than  half  of  T. 
For  ash — 

T  =  17,000  Ibs. ; 
C  =    9,000  Ibs. ;  and 
E  =  12,000  Ibs. ;  hence 
E  =  H  C  and  about  f  T. 

These  discrepancies  have  long  been  recognized,  and  the  cause 
has  generally  been  attributed  to  a  departure  from  the  law  of 
perfect  elasticity  and  a  movement  of  the  neutral  axis  away  from 
the  centre  of  the  beam  in  the  state  bordering  on  rupture  ;  but  as 
the  laws  of  these  variations  were  not  assigned,  their  influence 
could  not  be  analyzed.  (See  Articles  74  and  75.) 

The  tabulated  values  of  E  being  found  from  experiments  up- 
on solid  rectangular  beams,  they  are  especially  applicable  to  all 
beams  of  that  form,  and  they  answer  for  all  others  that  do  not 
depart  largely  from  that  form ;  but  if  they  depart  largely  from 
that  form,  as  in  the  case  of  the  J.  (double  T)  section,  or  hollow 
beams,  or  other  irregular  forms,  the  formulas  will  give  results 
somewhat  in  excess  of  the  true  strength ;  and  in  such  cases  Bar- 

O  7 

low's  theory  gives  results  more  nearly  correct. 

But  if,  instead  of  E,  we  use  T  or  C,  whichever  is  smaller,  in  the 
formulas  which  we  have  deduced,  and  suppose  that  the  neutral 


THE   RESISTANCE   OF  MATERIALS. 

axis  remains  at  the  centre  of  the  beam,  we  shall  always  oe  on 
the  safe  side  ;  but  there  would  often  be  an  excess  of  strength,  as, 
for  instance,  in  the  case  of  cast  iron  the  actual  strength  of  the 
beam  would  be  about  twice  as  strong  as  that  found  by  such  a 
computation. 

The  difficulty  is  avoided,  practically,  by  using  such  a  small 
fractional  part  of  R  as  that  it  will  be  considered  perfectly  safe. 
This  fraction  is  called  the  coefficient  of  safety.  The  values  com- 
monly used  for  beams  are  the  same  as  for  bars,  and  are  given 
in  Article  38. 

Experiments  should  be  made  upon  the  material  to  be  used  in 
a  structure,  in  order  to  determine  its  strength;  but  in  the  absence 
of  such  experiments  the  following  mean  values  of  R,  are 
used : — 

850  to  1,200  Ibs.  for  wood, 
10,000  to  15,000  Ibs.  for  wrought  iron,  and 
6,000  to  8,000  Ibs.  for  cast  iron. 


.     PRACTICAL,   FORMULAS. 

If  E  =  1,000  for  wood,  and 

12,000  for  wrought  iron, 

we  have  for  a  rectangular  beam,  supported  at  its  ends  and 
loaded  at  the  middle  of  its  length,    */^^p 

666  U*  t 
P  =  — Y. —  for  wooden  beams ;  and 

V 

_      8000  &? 

P  = j for  wrought-iron  beams. 

The  length  of  the  beam,  and  the  load  it  is  to  sustain,  are  gen- 
erally known  quantities,  and  the  breadth  and  depth  are  required ; 
but  it  is  also  necessary  to  assume  one  of  the  latter,  or  assign  a 
relation  between  them.  For  instance,  if  the  depth  be  n  times 
the  breadth,  the  preceding  formulas  give 


and  I  =  y — ;  and  d  =  y         ^   for  wrought  iron ;  (157) 


i* .  = 

2t 


,', 


TRANSVERSE  STRENGTH.  145 

122.  THE     RELATIVE     STRENGTH     OF  A    BEAM     Under 

the  various  conditions  that  it  is  held  is  as  the  moment  of  the 
applied  forces  ;  hence,  all  the  cases  which  have  been  con- 
sidered may,  relatively,  be  reduced  to  one,  by  finding  how  much 
a  beam  will  carry  which  is  fixed  at  one  end  and  loaded  at  the 
free  end,  equation  (146),  and  multiplying  the  results  by  the  fol- 
lowing factors  :  — 

FACTORS. 

Beam  fixed  at  one  end  and  loaded  at  the  other      -     -     -     1 
"  "  "  "     uniformly  loaded  -     2 

Beam  supported  at  its  ends  and  loaded  at  the  middle        -     4 
"  "  "          "     uniformly  loaded     -     -    -     8 

Beam  fixed  at  one  end  and  supported  at  the  other, 

and  uniformly  loaded      ...........     8 

Beam  fixed  at  both  ends  and  loaded  at  the  middle    -    -     8 
"  "     uniformly  loaded  -  12 

If  it  is  required  to  know  the  breadth  of  a  beam  which  will 
sustain  a  given  load,  find  &,  from  equation  (146)  ;  and  for  a  beam 
in  any  other  condition,  divide  by  the  factors  given  above  for 
the  corresponding  case. 

If  the  depth  is  required,  find  d  from  equation  (146),  and  di- 
vide the  result  for  the  particular  case  desired  by  the  square 
root  of  the  above  factors. 

123.  EXAMPI,ES.- 

1.  A  beam,  whose  depth  is  8  inches,  and  length  8  feet,  is  supported  at  its 
ends,  and  required  to  sustain  500  pounds  per  foot  of  its  length  ;  required 
its  breadth  so  that  it  will  have  a  factor  of  safety  of  -fa,  R  being  14,000 
pounds. 

From  equation  (146)  we  have, 

6PJ       6  x  500  x  8  x  8  x  12 

-        =^  niches; 


and  by  examining  the  above  table  of  factors  we  see  that  this  must  be  divided 
by  8  ;  •  '•  Ana.  =  3f4-  inches. 

2.  If  I  =  10  feet,  P  at  the  middle  =  2,000  Ibs.,  b  =  4  inches,  R  =  1,000  Ibs., 
required  d. 

3.  If  a  beam,  whose  length  is  8  feet,  breadth  ia  3  inches,  and  depth  6  inches, 

10 


146  THE   RESISTANCE    OF   MATERIALS. 

is  supported  at  its  ends,  and  is  broken  by  a  weight  of  10,000  pounds  placed  at 
the  middle,  and  the  weight  of  a  cubic  foot  of  the  beam  is  50  pounds ;  required 
the  value  of  R.  Use  equation  (150). 

4.  If  R  =  80,000  Ibs.,  I  =  12  feet,  b  =  2  inches,  d  —  5  inches,  how  much 
will  the  beam  sustain  if  supported  at  its  ends  and  loaded  uniformly  over  its 
whole  length,  coefficient  of  safety  i  ?  Ans  W  —  9,259  Ibs. 

5.  A  wooden  beam,  whose  length  is  12  feet,  is  supported  at  its  ends  ;  re- 
quired its  breadth  and  depth  so  that  it  shall  sustain  one  ton,  uniformly  distri- 
buted over  its  whole  length.     Let  R  =  15,000  Ibs.,  coefficient  of  safety  -jV,  and 
depth  =  4  times  the  breadth.  Ans.  b  =  2. 08  inches. 

d  =  8. 32  inches. 

6.  A  beam  is  2  inches  wide  and  8  inches  deep,  how  much  more  will  it  sustain 
with  its  broad  side  vertical,  than  with  it  horizontal  ? 

7.  A  wrought-iron  beam    12  feet  long,  2  inches  wide,  4  inches  deep,  is 
supported  at  its  ends.     The  material  weighs  i  Ib.  per  cubic  inch  ;  how  much 
load  will  it  sustain  uniformly  distributed  over  its  whole  length,  R  =  54,000 
Ibs.  ?  Ans.  Without  the  weight  of  the  beam,  15,712  Ibs. 

8.  A  beam  is    fixed  at  one  end ;    I  —  20   feet,  b  =  1-J  inch,  R  =  40,000 
Ibs. ;  weight  of  a  cubic  inch  of  the  beam  £  Ib.     Required  the  depth  that  it  may 
sustain  its  own  weight  and  500  Ibs.  at  the  free  end.  Ans.  4. 05  inches. 

9.  The  breadth  of  a  beam  is  3  inches,  depth  8  inches,  weight  of  a  cubic  foot 
of  the  beam  50  pounds,  R  r=  12,000;  required  the  length  so  that  the  beam 
shall  break  from  its  own  weight  when  supported  at  its  ends. 

Ans.  1  =  175. 27  feet. 

ISM.    RELATION    BETWEEN    STRAIN    AND    DEFLECTION. 

—When  the  strain  is  within  the  elastic  limit  we  may  easily  find 
the  greatest  strain  on  the  fibres  corresponding  to  a  given  deflec- 
tion. For  instance,  take  a  rectangular  beam,  supported  at  its 
ends  and  loaded  at  the  middle  of  its  length,  and  we  have  from 
equation  (148) 

f^J&P 

A^/'/jA 

and  from  equations  (T3)  and  (51)     A 

P/3 

A  =  J  ,-, .  ., ,  which  becomes,  by  substituting  P  from  the  preced- 
ing 


TRANSVERSE    STRENGTH. 


147 


(158) 


Examples.  —  1.  If  I  —  6  feet,  b  =  1£  inch,  d  =  4  inches,  coefficient  of  elas- 
ticity =  25,000,000  Ibs.  is  supported  at  its  ends  and  loaded  at  the  middle  so  as 
to  produce  a  deflection  at  the  middle  of  A  =  f  inch  ;  required  the  greatest 
strain  on  the  fibres.  Also  required  the  load. 

2.  On  the  same  beam,  if  the  greatest  strain  is  R  =  12,000  Ibs.,  required  the 
greatest  deflection. 

3.  If  the  beam  is  uniformly  loaded,  required  the  relation  between  the 
greatest  strain  and  the  greatest  deflection. 

.     4.  Generally,  prove  that  R  =  constant  x  —  A  . 

1S5.  HOLLOW  RECTANGUL.AR  BEAMS.  —  If  a  rectangular 
beam  has  a  rectangular  hollow,  both  symmetrically  placed  in 
reference  to  the  neutral  axis,  as  in  Fig.  62,  we  may 
find  its  strength  by  deducting  from,  the  strength 
of  a  solid  rectangular  beam  the  stroll  gtk  <>£  a  solid 

o  o 

beam  of  the  same  size  as  the  hollow.     But  in  this 

case,  when  the  beam  ruptures  at  #,  the  strain  at  1)' 

will  be  less  than  R.     As  the  strains  increase  di- 

rectly  as  the  distance  of  the  fibres  from  the  neutral  axis,  we  have, 

if  d  and  d'  are  the  depths  of  the  outside  and  hollow  parts  re- 

spectively,  g 

\d  :  \d'  ::  R  :  strain  at  I'  =  R  -T. 

T? 

If  lf  =  the  breadth  of  the  hollow,  the  stress  on  that  part,  if  it 
were  solid,  would  be,  according  to  equation  (145),  .JL 

jr  -ttj,* 

6 


Fjo-  62- 


which,  taken  from  equation  (145),  gives  for  the  resistance  of  a 
hollow  rectangular  beam,  p-^  fi£^  ^ 

bd3  —  b'd'3  7fcjt**&fr~fv-^  fcL.  *-  •>• 

fR,  -  -j—        -     -     -.    ,    .'-...     (159) 

" 


LI     =] 


If  the  hollow  be  on  the  outside,  as  in  Fig.  63, 
A      forming  an  II  section,  the  result  is  the  same.  , — I    I — , 

*  SH 

tvUJLtU^ey  f*4^,.   Fl0-<* 

' 
r   &.•  .  y  "  i  £    y 

/^t^e^uu  r^f-t  *%&&<&. 

p=  7&  i  'h*^  A   aJ~,~^,  ^^ 


148 


THE   RESISTANCE   OF   MATERIALS. 


126.  IF  THE  UPPER  AND  LOW  Ell  FLANGES  ARE  UN- 
EQUAL it  forms  a  double  T,  as  in  Fig.  64.  Let  the  notation 
be  as  in  the  figure,  and  also  di  equal  the  distance  from  the 
neutral  axis  to  the  upper  element,  and  x  the  distance  from  the 
neutral  axis  to  the  lower  element. 


FIG.  64. 

To  find  the  position  of  the  neutral  axis,  make  the  statical 
moments  of  the  surface  above  it  equal  to  those  below  it.  This 
gives 

d'  V  (d,  -  i  d')  +  i  I"'  (d,  -  dj  =--  d"  I"  (a?  -  J  d")  +  i  I'" 


We  also  have  d,  =  d  -  te  =  d'  <f  d"  +  d"'  -  x         (161) 

These  equations  will  give  x  and  dv. 

Constructing  the  wedges  as  before,  and  the  resistance  to  com- 
pression is  represented  by  the  wedge  whose  base  is  V  dl  and 
altitude  R,  minus  the  wedge  whose  base  is  (&'  —  ~b'"\  (dl  —  df) 

d  —d' 
and  altitude  -—^  —  R.     Hence  the  resistance  to  compression  is 


-   E  (V  -  V")  (dt  -  d') 


/  _ 


TRANSVERSE  STRENGTH.  149 

The  centre  of  gravity  is  at  f  the  altitude,  or  \dv  for  the  for- 
mer wedge,  and  %(d1  —  d')  for  the  latter,  and  if  the  volumes 
be  multiplied  by  these  quantities  respectively,  it  will  give  for 
the  moment  of  resistance  to  compression 

\Kh'd?  -  ^  (bf  -  V")  (d,  -  dj 

CL/I 

Next  consider  the  resistance  to  tension.  Since  the  strains 
on  the  elements  are  proportional  to  their  distances  from  the  neu- 
tral axis,  therefore 

T> 

d^  :  x  :  :  R,  :  strain  at  the  lower  side  of  the  section  =  —  r  #, 
and  similarly, 
d1  :  (x  —  d")  :  :  R  :   strain  at  the  opposite  side  of  the  lower 

R 

flange  =  -^-(x  —d"\ 

Hence  the  tensive  strains  will  be  represented  by  a  wedge  whose 

base  is  Wx  and  altitude  •T-a?,  minus  a  wedge  whose  base  is(V'  '— 

>'  R 

and  altitude  -y  (x  —  \d").  Hence  the  moment  of  resistance  is 


-J      (V  -  V")  (x  -  d'^f 

istance  is  the  su 

J  I}'  <*''  ~  (Z/  ~  J///)  (dl  ~  d'^ 


The  total  moment  of  resistance  is  the  sum  of  the  two  moments,  or 
'     '  ~        ~    ///          ~     '  //  /// 


.    -    .       (162) 

For  a  single  T  make  £>"  and  d"  =  0  in  the  above  expression. 

The  method  which  has  here  been  applied  to  rectangular 
beams  may  be  applied  to  beams  of  any  form  ;  but  it  often  re- 
quires a  knowledge  of  higher  mathematics  to  find  the  volume  of 
the  wedge,  and  the  position  of  its  centre  of  gravity  ;  or  resort 
must  be  had  to  ingenious  methods  in  connection  with  actual 
wedges  of  similar  dimensions. 

6  ------ 

:  '  *£x  --  /X  /&U  ^)6  -i^f  ti/£~i^(' 

^-^-•X>V*  yL^ 

. 


150 


THE   RESISTANCE    OF  MATERIALS. 


TRUE    VALUE    OF    ^     AND    AN    EXAMPLE. — In  this 

and  similar  expressions 

d^  =  the  distance  from  the  neutral  axis  to  the  fibre  most  re- 
mote from  it  ON  THE  SIDE  WHICH  FIRST  RUPTURES. 

dl  is  usually  taken  as  the  distance  to  the  most  remote  fibre, 
without  considering  whether  rupture  will  take  place  on  that 
side  or  not ;  but  this  oversight  may  lead  to  large  errors. 

For  example,  let  the  dimensions  of  a  cast- 
iron  double  T  beam  be  as  in  Fig.  65,  and  228 
inches  between  the  supports.  Required  the 
load  at  the  middle  necessary  to  break  it. 

The  position  of  the  neutral  axis  is  found 
from  equations  (160)  and  (161)  to  be  7.96 
inches  from  the  lower  side,  and  11.54  inches 
from  the  upper.  As  cast  iron  will  resist  from 
four  to  six  times  as  much  ^compression  as>Kjr 
tension — this  beam  will  rupture  on  the  lower 
side  first ;  hence  d^  in  the  equation  =  7.96  inches.  As  the 
value  of  R  is  not  known,  take  a  mean  value  —  36,000  Ibs.  The 
moment  of  the  rupturing  force — neglecting  the  weight  of  the 
beam — is  J  P^,  which  placed  equal  to  expression  (162)  and  re- 
duced gives 


FIG.  65. 


p  — 


4 

228 


36,000 
7.96 


X  1,672  =  132,030  Ibs.  =58.9  tons  gross. 


Had  we  used  dt  =  11.54,  it  would  have  given  P  =  40.4  tons. 
Such  beams  actually  broke  with  from  50  to  54  tons ;  or,  in- 
cluding the  weight  of  the  beam,  with  a  mean  value  of  52J  tons. 

By  reversing  the  problem,  and  using  52^  tons  for  P,  we  find 
that  R  is  a  little  more  than  32,000  pounds.  Had  this  value  of  R 
been  used  in  the  first  solution,  and  dl  made  equal  11.54,  it  would 
have  given  for  P  a  little  more  than  36  tons,  which  would  be  the 
strength  if  the  beam  were  inverted.  If  the  upper  flange  were 
smaller  or  the  lower  larger,  the  discrepancy  would  have  been 
greater. 

The  strain  upon  a  fibre  in  thejogper  surface  is  to  the  strain  upon 
one  in  the  lower  surface  as[^a  to  #}  hence,  if  the  material  resists 
more  to  compression  than  to  tension  (as  cast  iron),  it  should 
be  so  placed  that  the  small  flange  shall  resist  the  former,  and 


sltjt^  iHrtvetA/L 


TRANSVERSE    STRENGTH. 


151 


the  large  one  the  latter.  If  a  cast-iron  beam  be  supported  at 
its  ends,  the  smaller  flange  should  be  uppermost,  and  as  it  re- 
sists from  four  to  six  times  as  much  compression  as  tension,  the 
neutral  axis  should  be  from  four  to  six  times  as  far  from 
the  upper  surface  as  from  the  lower,  for  economy.  Using  the 
same  notation  as  in  Fig.  64  and  we  have, 

d,       greatest  compressive  strain  yt&r^  4*  • 

x  ~         greatest  tensive  strain 
and  for  economy  we  should  have, 

dl        ultimate  compressive  strength 
x  ~       ultimate  tensile  strength 

The  ultimate  resistance  of  wrought  iron  is  greater  for  ten- 
sion than  for  compression ;  hence,  if  a  wrought-iron  beam  is 
supported  at  its  ends,  the  heavier  flange  should  be  uppermost. 

The  proper  thickness  of  the  vertical  web  can  be  determined 
only  by  experiment,  and  this  has  been  done,  in  a  measure,  by 
Baron  von  Weber,  in  his  experiments  on  permanent  way. 


^_ 

34f 


EXPERIMENTS  OF  BARON  VON  WERER  for  deter- 
mining the  thickness  required  for  the  central  web  of  rails. 

Baron,  von  Weber  desired  to  ascertain  what  was  the  mini- 
mum thickness  which  could  be  given  to  the  web  of  a  rail,  in 
order  that  the  latter  might  still  possess  a  greater  power  of  re- 
sistance to  lateral  forces  than  the  fastenings  by  which  it  was 


FIG.  65a. 


secured  to  the  sleepers.  For  this  purpose  a  piece  of  rail  6  feet 
in  length,  rolled,  of  the  best  iron  at  the  Laurahutte,  in  Silesia, 
was  supported  at  distances  of  35.43  in.,  and  loaded  nearly  to  the 
limit  of  elasticity  (which  had  been  determined  previously  b  ex- 


152 


THE    RESISTANCE    OF    MATERIALS. 


periments  on  other  pieces  of  the  same  rail),  and  the  deflections 
were  then  measured  with  great  care  by  an  instrument  capable 
of  registering  1-1000  in.  with  accuracy.  This  having  been  done, 
the  web  of  the  piece  of  rail  was  planed  down,  and  each  time 
that  the  thickness  had  been  reduced  3  millimetres  the  vertical 
deflection  of  the  rail  under  the  above  load  was  again  tested,  and 
the  rail  was  subjected  to  the  following  rough  but  practical  ex- 
periments. The  piece  of  rail  was  fastened  to  twice  as  many  fir 
sleepers  by  double  the  number  of  spikes  which  would  be  em- 
ployed in  practice,  and  a  lateral  pressure  was  then  applied  to  the 
head  of  the  rail  by  means  of  a  lifting-jack,  until  the  rail  began 
to  cant  and  the  spikes  were  drawn.  The  same  thing  was  then 
done  by  a  sudden  pull,  the  apparatus  used  being  a  long  lever  fas- 
tened to  the  top  of  the  rail,  as  shown  in  Fig.  65a.  The  lifting- 
jack  and  the  lever  were  applied  to  the  ends  of  the  rail,  and  the 


FIG.  65b. 

web  of  the  latter  had,  in  each  case,  to  resist  the  whole  strain  re- 
quired for  drawing  out  the  spikes.  The  results  of  the  experi- 
ments made  to  ascertain  the  resistance  of  the  rail  to  vertical 
flexure  with  different  thicknesses  of  web,  and  under  a  load  of 
5,000  Ibs.,  were  as  follows : — 


TRANSVERSE    STRENGTH. 

Thickness  of  web.  Vertical  deflection. 

In.  In. 

15  millimetres  =  0.59 0.016 

12          "               0.47    -    -  0.016 

9          "               0.35     -    -    -    -  0.019 

6          "               0.24    -    -         -    -  0.0194 

3          «               0.12 0.022 


153 


These  results  showed  ample  stiffness,  even  when  the  web  was 
reduced  in  thickness  to  0.12  in.  To  determine  the  power  of 
resistance  of  the  rail  to  lateral  flexure,  an  impression  of  the  sec- 
tion was  taken  in  lead  each  time  that  the  spikes  were  drawn. 

The  forces  applied  in  these  experiments  were  very  far  greater 
than  those  occurring  in  practice,  yet  it  was  found  that  with  the 
web  12,  9,  and  even  6  millimetres  thick,  no  distortion  took  place, 
and  only  when  the  thickness  of  the  web  was  reduced  to  3  milli- 
metres (0.12  in.)  was  a  slight  permanent  lateral  deflection  of  the 
head  caused  just  as  the  spikes  gave  way.  The  section  shown  in 
Fig.  65b  had  then  been  reduced  to  that  shown  in  Fig.  65c. 


985-i-i4 


FIG.  65c. 


Next,  a  rail,  with  the  web  reduced  to  3  mill.  (0.12  in.)  in 
thickness,  was  placed  in  the  line  leading  to  a  turn-table  on  the 


154  THE   RESISTANCE   OF   MATERIALS. 

Western  Railway  of  Saxony,  where  it  has  remained  until  the 
present  time,  1870,  receiving  the  shocks  due  to  engines  passing 
to  and  from  the  turn-table  more  than  one  hundred  times  daily. 
It  follows  from  these  experiments  that  the  least  thickness  ever 
given  to  the  webs  of  rails  in  practice  is  more  than  sufficient,  and 
that  if  it  were  possible  to  roll  webs  J  in.  thick,  such  webs  would 
be  amply  strong,  if  it  were  not  that  there  would  be  a  chance  of 
their  being  torn  at  the  points  where  they  are  traversed  by  the 
fish-plate  bolts.  Baron  von  Weber  concludes  that  webs  f  in.  or 
|-  in.  thick  are  amply  strong  enough  for  rails  of  any  ordinary 
height,  and  that,  in  fact,  the  webs  should  be  made  as  thin  as  the 
process  of  rolling  and  as  the  provision  of  sufficient  bearing  for 
the  fish-plate  bolts  will  permit. 

]  29.  ANOTHER  GRAPHICAL  METHOD — If  manipulating 
processes  are  to  be  used  for  determining  the  strength,  the  fol- 
lowing method  possesses  many  advantages  over  the  former. 

Since  the  strains  vary  directly  as  their  dis- 
tance from  the  neutral  axis,  the  triangle 
ABC  (Fig.  66),  in  the  rectangle  BODE, 
represents  the  compressive  strains  if  each 
element  of  the  shaded  part  has  a  strain 
equal  to  R;  and  its  moment  is  R  times 
the  area  multiplied  by  the  distance  of  the 
centre  of  gravity  of  the  triaiip-le  from  the 

FIG.  66.  -,         . 

neutral  axis ;  or, 

R  x  (t>  x  %  of  $d)  x  f  of  %d  =  •&!&€?, 

and  the  moment  of  tensile  resistance  is  the  same,  hence  the 
total  moment  is  double  this,  or  ^Rbd*,  as  found  by  the  preced- 
ing process. 

13O.     IF  A  SQUARE  BEAM  HAVE  ONE  OF  ITS  DIAGONALS 

VERTICAL  (Fig.  67),  the  neutral  axis  will  coincide  with  the 
other  diagonal.  Take  any  element,  as  ab,  and  project  it  on  a  line 
cd,  which  passes  through  A  and  is  parallel  to  BC,  and  draw  the 
lines  Oc  and  Od,  and  note  the  points  f  and  g  where  they  intersect 
the  line  ab.  If  the  element  were  at  cd,  the  strain  upon  it  would 
be  R,  multiplied  by  the  area  of  cd,  or  simply  H.cd ;  but  because 


TRANSVERSE    STRENGTH. 


155 


the  strains  are  directly  proportional  to  the  distances  of  the  ele- 
ments from  the  neutral  axis,  the  strain  on  ab  is  1&.fg.  Proceed 
in  this  way  with  all  the  elements  and  construct  the  shaded 
figure.  The  strains  on  the  upper  part  of  the  figure  ABC,  which 


FIG.  67J 


begin  with  zero  at  BC,  and  increase  gradually  to  R,  at  A,  will  be 
equivalent  to  the  strains  on  the  shaded  figure  AO,  if  the  strain 
is  equal  to  R  on  each  unit  of  its  surface.  Hence  the  total  strain 
on  each  half  is  the  area  of  the  shaded  part  AO,  multiplied  by 
R,  and  the  moment  of  the  strain  of  each  part  is  this  product 
multiplied  by  the  distance  of  the  centre  of  the  shaded  part  from 
the  axis  BC. 

By  similar  triangles  we  have 

Aa  :  ab  :  :  AB  :  BC,  and 
cd=  ab  :fg  :  :  AO  :  x  :  :  AB  :  Ba  or  AB  —  Aa  ; 

x  being  the  distance  of  fg  from  O. 
From  these  eliminate  ab,  and  find 

BC 

a/=  %(ab  -fg)  = 


hence  the  curve  which  bounds  the  shaded  figure  is  a  parabola 
which  is  tangent  to  AB,  and  whose  axis  is  parallel  to  BC. 
Let  d  =  one  side  of  the  square,  then 


=  AO,  and 

i.  |/2<tf  =  the  widest  part  of  the  shaded  figure. 
The  area  of  a  parabola  is  two-thirds  the  area  of  a  circumscribed 
rectangle. 


156  THE   RESISTANCE    OF   MATERIALS. 

Hence  the  area  of  AO  is 
and  the  moment  is 


~  12  |/2 ' 
and  the  moment  of  both  sides,  multiplied  by  R,  is 

R^ 


(163) 


If  b  —  d  in  equation  (145)  and  the  result  compared  with  the 
above,  we  find  :— 

The  strength  of  a  square  beam  with  its  side  vertical  :  strength 
of  the  same  beam  with  one  of  its  diagonals  vertical  :  :  i/2  :  1 
or  as  7  :  5  nearly. 

So  that  increased  depth  merely  is  not  a  sufficient  guarantee 
of  increased  strength.  The  reason  why  the  strength  is  dimin- 
ished when  the  diagonal  is  vertical,  is  because  there  is  a  very 
small  area  at  the  vertex  where  the  strain  is  greatest,  but  when 
a  side  is  horizontal  the  whole  width  resists  the  maximum  strain. 

131.  IRREGULAR  SECTIONS. — This  method  is  applicable 
to  irregular  sections,  as  shown  by  the  following  example. 


FIG.  68. 


Let  Fig.  68  be  a  cross  section  of  a  beam.     In  a  practical 
case  it  may  be  well  to  make  an  exact  pattern  of  the  cross 


TRANSVERSE    STRENGTH.  157 

section,  of  stiff  paper  or  of  a  thin  board  of  uniform  thick- 
ness. To  find  the  position  of  the  neutral  axis,  draw  a  line 
on  the  pattern  which  shall  be  perpendicular  to  the  direc- 
tion of  the  forces  which  act  upon  the  beam,  that  is,  if  the 
forces  are  vertical  the  line  will  be  horizontal.  In  a  form 
like  Fig.  68  this  line  will  naturally  be  parallel  to  the  base  of 
the  figure.  Then  balance  the  pattern  on  a  knife-edge,  keeping 
the  base  of  the  figure  (or  the  line  previously  drawn)  parallel  to 
the  knife-edge,  and  when  it  is  balanced  the  line  of  support  will 
be  the  neutral  axis.  Proceed  to  construct  the  shaded  part  as 
shown  in  the  figure,  by  projecting  any  element,  as  ab  on  the  line 
cd,  and  drawing  cO  and  dO,  and  noting  the  intersections  f  and  g, 
the  same  as  in  Fig.  67.  The  elements  on  the  lower  side  must 
be  projected  on  a  line  mn,  which  is  at  the  same  distance  from 
the  neutral  axis  as  the  most  remote  element  on  the  upper  side. 
The  area  of  the  shaded  part  above  the  neutral  axis  should  equal 
that  below,  because  the  resistance  to  extension  equals  that  for 
compression.  The  area  of  the  shaded  part  may  be  found  ap- 
proximately by  dividing  it  into  small  rectangles  of  known  size, 
and  adding  together  the  full  rectangles  and  estimating  the  sum 
of  the  fractional  parts.  Or,  the  shaded  part  may  be  cut  out  and 
carefully  weighed  or  balanced  by  a  rectangle  of  the  same  mate- 
rial, after  which  the  sides  of  the  rectangle  may  be  carefully 
measured  and  contents  computed.  The  area  of  the  rectangle 
would  evidently  equal  the  area  of  the  irregular  figure. 

The  ordinate  to  the  centre  of  gravity  of  each  part  may  be 
determined  by  cutting  out  the  shaded  parts  and  balancing  each 
of  them  separately  on  a  knife-edge,  as  before  explained,  keeping 
the  knife-edge  parallel  to  the  neutral  axis.  The  distance  be- 
tween the  line  of  support  and  the  neutral  axis  will  be  the  ordi- 
nate to  the  centre  of  gravity.  The  moment  of  resistance  is  then 
found  by  multiplying  the  area  of  each  shaded  part  by  the  dis- 
tance of  its  centre  of  gravity  from  the  neutral  axis,  and  multi- 
plying the  siim  of  the  products  ~by  R. 

These  mechanical  methods  may  be  managed  by  persons  who 
have  only  a  very  limited  knowledge  of  mathematics,  and  if 
skilfully  and  carefully  done  will  give  satisfactory  results.  It 
does  not,  however,  furnish  such  an  uniform,  direct  and  exa-ct 
mode  of  solution  as  the  analytical  method  which  is  hereafter 
explained. 


158 


THE   RESISTANCE    OF   MATERIALS. 


FORMULA  OF  STRENGTH  ACCORDING  TO  BARLOW'S 

THEORY.-Either  of  the  above  methods  may  be  used.  One 
part  of  the  expression  for  the  strength  is  of  the  same  form  as 
that  found  by  the  common  theory  ;  but  instead  of  R  we  must  use 
T,  or  C  —  the  former  if  it  ruptures  by  tension,  the  latter  if  by 
crushing.  The  other  resistance,  <£,  for  solid  beams  is  evenly  dis- 
tributed over  the  surface.  For  example,  take  a  rectangular 
beam,  Fig.  61,  and  the  resistance  to  longitudinal  shearing  on  the 
upper  side  is  <f>  1}  x  \d  =  J  <£  bd,  and  its  moment  is  -J-  (f>  bd  x  $ 
of  %d  —  -J-  <£  bd*,  and  for  both  sides,  J  <f>  bd?.  Hence,  according 
to  Barlow's  theory,  the  expression  for  the  strength  of  a  rectan- 
gular beam  is 

[  J  $  +  £T]  bd*  for  cast  iron,  and 

[i  $  +  i^]  bd*  for  wrought  iron  and  wood    -     -     (164) 
If  the  beam  is  supported  at  its  ends  and  loaded  at  the  middle, 
we  have 


=  [J  <f>  +  -J-T]  bd?  for  cast  iron  -     -    -    -          (165) 

The  volume  which  represents  the  resistance  due  to  <f>  is 
always  a  prism,  having  for  its  base  the  surface  of  the  figure  and 
$,  or  some  fraction  of  <£,  for  its  altitude.  If  the  second  method 
of  illustration  be  used,  it  will  take  two  figures  to  fully  illustrate 
the  strains.  For  instance,  if  the  section  be  as  in  Fig.  68,  the 
moment  of  the  shaded  part  will  be  multiplied  by  T  or  C,  as  the 
case  may  be.  To  find  the  remaining  part  of  the  moment,  find 
the  area  of  each  part  of  the  transverse  section,  also  the  distance 
of  the  centre  of  gravity  of  each  part  from  the  neutral  axis. 
Then,  to  find  the  moment  of  re- 
sistance due  to  longitudinal  shear- 
ing, multiply  the  area  of  each  part 
by  the  distance  of  its  centre  of  grav- 
ity from  the  neutral  axis,  add  the 
products  and  multiply  the  sum  by  <f>. 
This  is  true  for  solid  sections  ;  but 
for  hollow  beams,  T  and  H  sections, 
where  there  is  an  abrupt  angular 
change  from  the  flange  to  the  verti- 
cal part  of  the  beam,  the  factor  <£  requires  a  modification.  For 
instance,  take  the  simple  case  of  a  single  T,  Fig.  69,  in  which 


TUANS  VERSE    STRENGTH.  159 

the  breadth  of  the  T  is  b'  and  its  depth  d  ',  and  the  other  nota- 
tion as  in  the  figure. 

The  resistance  of  the  upper  part  is  represented  by  the  prism 
whose  base  is  bx,  and  whose  altitude  is  <£,  plus  the  prism  whose 

base  is  d'  (b'-b),  and  whose  altitude  is  —  <£.  The  resistance  of 
the  lower  part  is  $  bdr  The  total  moment  of  this  resistance  is  — 
+  d'(V-b)  x  -  $  (x  - 


To  this  add  the  moment  of  resistance  for  direct  extension 
and  compression,  the  expression  for  which  is  of  the  same  form 
as  for  common  theory,  and  we  have  for  the  total  moment  :  — 

—  ;  (/>(&'-  b)  (X-  %d')  +  £<£    bdS  +  Q^r  \])d:+  VX*-(V-  b) 


(x-dy\     ..............    (166) 

From    numerous    experiments  made   upon   cast-iron  beams 
having  a  variety  of  cross  sections,  Barlow  found  that  $  varied 
nearly  as  T,  that  practically  it  was  a  fraction  of  T,  the  mean 
value  of  which  was  0.9T. 
For  wrought  iron  he  found  $  =  0.53T 

=  0.8C  nearly. 

Peter  Barlow,  F.E.S.,  father  of  W.  H.  Barlow,  F.R.S.,  the 
latter  of  whom  proposed  the  "  theory  of  flexure,"  in  an  article 
in  the  Civ.  Eng.  Jour.,  Yol.  xxi.,  p.  113,  assumes  that  <£  =  T. 

From  the  above  it  is  inferred  that  the  practical  mean  values 
of  <f)  are  :  — 

16,000  Ibs.  for  cast  iron. 
30,000  Ibs.  for  wrought  iron. 
8,000  Ibs.  for  wood. 

Examples.—  I.  How  much  will  a  beam  whose  length  is  12  feet,  breadth  2 
inches,  depth  o  inches,  sustain,  if  supported  at  its  ends,  and  uniformly  loaded 
over  its  whole  length,  and  C  =  50,000  Ibs.,  <p  =  30,000  Ibs.,  and  coefficient  of 
safety  ±  ?  Ans.—  11,000  Ibs.  nearly. 

—2.  If  0  =  T  =  16,000  Ibs.,  b  =  2  inches,  d  =  5  inches,  I  =  8  feet  ;  required 
the  uniform  load  which  it  will  sustain  with  a  coefficient  of  safety  of  £. 

""U.  Itb  =  d  =  2  inches,  1=6  feet,   R=  50,000  Ibs.,  is  broken  by  an  uni- 
form load  of  10,000  pounds,  required  0. 


BEAMS     LOADED  g  AT     ANY     NTJUIBER    OF  POINTS.  - 

If  the  beam  is  loaded  otherwise  than  has  heretofore  been  sup- 
posed, it  is  only  necessary  to  find  the  moment  of  all  the  forces 


160 


THE   RESISTANCE   OF   MATERIALS. 


in  reference  to  the  centre  of  a  section  and  place  the  algebraio 
sum  equal  to  the  moments  of  resistance.  Those  which  act  in 
opposite  directions  will  have  contrary  signs. 

For  instance,  if  a  beam,  AB,  Fig.  70,  rests  upon  two  supports, 
and  has  weights,  P1?  P2, 

P3,  &c.,  resting  upon  it  at       v,  v2 

distances  respectively  of 
nu  nv  nv  &c->  from  one 
support,  and  m,,  m,,  ma, 
&c.,  from  the  other,  the 
sum  of  the  moments  of 
the  forces  on  any  section 
C  whose  distance  is  x 
from  the  support  A,  is 

Y™   "P    (™    m    \  T>    (M  ^    \ 

JB  —  r^x  —nj  —  r^(x  —  nj 

P  in  which  n  is  less  than  x. 
gular  beams. 

Y1?  the  reaction  of  one  support,  is  readily  found  by  taking 
the  moments  of  all  the  external  forces  about  B,  and  solving  for 
Y,,  thus :— 


Fia.  70. 

<fec.,  to  include  all]  the  terms  of 
This  equals  ^Rbd*  for  rectan- 


Similarly  Y2  = 

also,  Y,  +  Ya  =P,  +  P2  +  P3  +  &c.  =  5  P 


v 


134.    A  PARTIAL,  UNIFORM  LOAD. 

uniformly  over  any  portion  of  its 
length,  as  in  Fig.  71. 

Let  I  =  AB  —  length  of  beam ; 
2a  =  DE  =  length  of   the 

uniform  load ; 
x  =  AF  =  the  distance  to 

any  section ; 
w  =  the  load  on  a  unit  of  length  ; 
Y  =  the  reaction  of  the  support  A  ; 
C  the  centre  of  the  load  ; 
I,  =  AC  ;  I  =CP>. 


-Let  the  beam  be  loaded 


.../„.. 


FT  

\£  —  -^2 

M. 

~*     Cl 
F 

FJG.  71. 

M 

-   9  A 


TRANSVERSE  STRENGTH.   •C'  161 

Then  AD  =  1,-a,  and  DF  =  x  -  I,  +  a. 
Load  on  DF  =  w  (x  —  I,  +  a), 


By  the  principle  of  moments 


The  moment  of  stress  at  F  is 


:  m  (167)       ^ 

\L~  £  +•- 

That  value  of  a?  which  will  make  equation  (167)  a  maximum, 
gives  the  position  of  the  dangerous  section.    Differentiate,  place  y  *. 
equal  zero,  and  make  ll  +  1^  =  I,  and  solve  for  a?,  and  find     //^ 

-     -^    .^7   <168) 


^  *-  -t*  +  >3--2> 

A  a     '  '      '  M 

l !  <  \l->  &  >  ^i ; 

so  that  the  maximum  strain  is  at  the  centre  of  the  loading  only 
when  the  centre  of  the  loading  is  over  the  centre  of  the  beam  ; 
and  in  all  other  cases  it  is  nearer  the  centre  of  the  beam  than  the 
centre  of  the  loading  is. 

The  maximum  strain  is  found  by  substituting  the  value  of  x 
equation  (168)  in  equation  (167). 

The  following  interesting  facts  are  also  proved. 

Let  AD  =  y  /.  a  =  ll  —  y  which  in  equation  (168)  reduces  it  to 

•^ 

which  is  a  maximum  for  y  =  0  ;  hence  so  far  as  A  D  is  con- 
cerned, equation  (168#)  is  a  maximum  when  one  end  of  the  load  , 
is  over  the  support,  and  for  this  case  the  equation  becomes  V^V 


• 

which  is  a  maximum  for  ^  =  J  I  or  2Zt  =  £Z,  or  the  load  must 
11 


162 


THE   RESISTANCE   OF   MATERIALS. 


extend  to  the  middle  of  the  beam. 
equation  (168)  becomes 


Making  a  =  Z,  =  J  I,  and 


and  these  values  of  Z,  and  x  in  equation  (167)  give  for  the  max- 
imum moment  of  stress, 


(169) 


in  which  W  is  the  load  on  half  the  beam. 

Equation  (167)  gives  the  stress  at  the  middle  of  the  load,  by 
making  a  —  l^  —  \l  and  x  =  J  I.  This  gives  -J  WZ  f  or  the  stress 
at  the  middle  of  the  loading  ;  hence,  the  maximum  stress  is  1-J 
times  the  stress  at  the  middle  of  the  loading  when  the  load 
extends  from  the  one  support  to  the  middle  of  the  beam. 


OBLIQUE  STRAINS.  —  If  the  force  be  inclined  to  the 
axis,  as  in  Figs.  72  and  73,  let  6  =  the  angle  which  P  makes 
with  a  normal  section. 


(f*r, 


FIG.  72. 


FIG.  73. 


Then,  P  cos  6  =  normal  component, 

P  sin  6  =  longitudinal  component. 

If  K  =  the  transverse  section,  then 

P  sin  6 

— =~—  =  the  tension  or  compression  upon  a  unit  of  sec- 
tion which  arises  directly  from  the  longitudinal  component. 
This  tends  directly  to  dimmish  R  in  the  formula  whether  6  be 
obtuse  or  acute.  If  the  beam  be  fixed  at  one  end  and  free  at 
the  other,  as  in  Fig.  72,  the  equation  of  moments  becomes : — 


TRANSVERSE  STRENGTH.  163 

which  for  rectangular  beams  becomes 

(170) 


136.  GENERAL  FORMULA.  —  The  preceding  methods  are 
easily  understood,  and  are  perhaps  sufficient  for  the  more  simple 
cases  ;  but  for  the  purposes  of  analysis  a  general  formula  is  bet- 
ter, by  means  of  which  a  direct  analytical  solution  may  be 
made  for  special  cases. 

Let  E,  —  the  'modulus  of  rupture  ^  as  explained  in  article  120  ; 
x  and  u  horizontal  coordinate  axes,  the  former  coinciding 

with  the  axis  of  the  beam,  and  y  a  vertical  axis  ; 
Then  B-  du  dy  =  the  resistance  of  a  fibre  which  is  most  remote 

from  the  neutral  axis. 
Let  dl  =  distance  between  the  neutral  axis  and  the  most  remote 

fibre  ;  then,  according  to  the  common  theory,  since 

the  strains  vary  as  the  distance  from  the  neutral  axis 

di  :  y  :  :  ~Rdudy  :  resistance  of  any  fibre  =  -j  y  dy  du 

#, 

T> 

/.  -7  y2  dydu  =  the  moment  of  resistance  of  any  fibre, 

#! 

and  the  sum  of  all  the  moments  of  resistance  of  any  section  is 


C     C 

J  J 


r 

l 

which  is  called  the  moment  of  rupture,  and  must  equal  the 
sum  of  the  moments  of  straining  forces  ; 

.:SPx=*I         .........  (171) 

The  second  member  of  this  equation  involves  the  character  of 

the  material  (E)  and  the  form  of  the  transverse  sections  (  v)  ;  the 

#t 

latter  of  which  may  be  determined  by  analysis,  and  the  former 
by  experiment.  The  second  member  shows  that  for  economy 
the  material  should  be  removed  as  much  as  possible  from  the 
neutral  axis.  A  few  special  cases  will  now  be  given, 


-S«  -        '     *+-**—  J>/ 


164 


THE   RESISTANCE    OF   MATERIALS. 


137.  I'ET  THE  BEAM  BE  RECTANGULAR,  5  the  breadth, 
and  d  the  depth,  as  in  Fig.  61, 


Then  I  = 


T> 


-•/T 

*/0         e/O 


u  = 


/.  -v  I  =  -J-  R  J<#2  which  is  the  same  as  expression  (145). 


138.     IF  THE  SIDES  OF  THE  BEAM  ARE  INCLINED    to   the 


FIG.  74. 


FIG.  75. 


direction  of  the  force,  as  in  Fig.  74,  let  i  be  the  inclination  of 
the  side  to  the  horizontal  ;  then 


'S'  1  , 

—  I  -    -    -    -     (17-i) 
a  J 


dl  = 


This  expression  has  an  algebraic  minimum, f  but  not  an 
braic  maximum.  By  inspection,  however,  we  find  that  the 
practical  maximum  is  found  by  making  i  ~  90°,  if  d  exceeds  l>. 
Hence,  a  rectangular  beam  is  strongest  when  its  broad  side  is 
parallel  to  the  direction  of  the  applied  forces. 

Hence,  the  braces  between  joists  in  flooring,  as  in  Fig.  75,  not 

*  See  Appendix  III. 

f  See  an  article  by  the  author  in  the  Journal  of  Franklin  Institute,  Vol. 
LXXV.,p.  2GO. 


r*. 


TRANSVERSE  STRENGTH.  165 

only  serve  to  transmit  the  stresses  from  one  to  another,  but 
also  to  strengthen  them  by  keeping  the  sides  vertical. 
If  i  =  90°,  equation  (172)  becomes  $Rbd*  -     -    -     (173) 
If  b  =  d  and  i  =  45°,  equation  (172)  reduces  to 


62     - 

(which  is  the  same  as  equation  (163)), 
and  ifb  =  d,  and  i  =  0°  or  90°,  it  becomes 


Hence,  the  strength  of  a  square  beam  having  a  side  vertical 
is  to  the  strength  of  the  same  beam  having  its  diagonal  vertical, 
as 


or  -/2  to  1  or  as  7  to  5  nearly, 

In  establishing  equation  (172)  it  was  assumed  that  the  neutral 
surface  was  perpendicular  to  the  direction  of  the  applied  forces, 
which  is  not  strictly  true  unless  the  forces  coincide  with  the 
diagonal  ;  for  in  other  cases  there  is  a  stronger  tendency  to 
deflect  sidewise  than  in  the  direction  of  the  depth.  In  this  case, 
as  soon  as  the  beam  is  bent  there  is  a  tendency  to  torsion.  Both 
these  conditions  make  the  beam  weaker  than  when  the  sides  are 
vertical.  If  the  tendency  to  torsion  be  neglected,  the  case  may 
be  easily  solved  ;  but  as  the  result  shows  the  advantage  of  keep- 
ing the  sides  vertical,  the  solution  is  omitted. 

1  39.     THE    STRONGEST    RECTANGULAR  BEAM  which  Can 

be  cut  from  a  cylindrical  one  has  the  breadth 
to  the  depth  as  1  to  \/  2,  or  nearly  as  5  to  7. 
Let  x  —  AB  —  the  breadth, 
y  =  AC  =  the  depth,  and 

I)  =  AD  =  the  diameter. 

—  —  ^  jf 

\.  76. 

Then, 

Sf  =  D'-3- 

and  equation  (173)  becomes 


166 


THE   RESISTANCE    OF   MATERIALS. 


which  by  the  Differential  Calculus  is  found  to  be  a  maximum 
for 


.*.  x  :  y  : :  1  :  -y/2  or  nearly  as  5  to  7. 

Examples. — How  much  stronger  is  a  cylindrical  beam  than  ihe  strongest 
rectangular  one  which  can  be  cut  from  it  ? 
(For  the  strength  of  a  cylindrical  beam,  see  equation  (180)). 

-*.±KTr  ^  Ans-A] 

How  much  stronger  is  the  strongest  rectangular  beam  that  can  be  cut  from 
a  cylindrical  one,  than  the  greatest  square  beam  which  can  be  cut  from  it  ? 


cent- 


14O.  TRIANGULAR  REAMS. — If  the  base  is  perpendicular 
to  the  neutral  axis,  as  in  Fig.  77 ; 

Let  d  =  AD  =  the  altitude,  and 
I  =  BC  =  the  base. 

Take  the  origin  of  coordinates  at  the  cen- 
tre of  gravity  of  the  triangle,  y  vertical  and 
u  horizontal. 

Then,  by  similar  triangles, 

\db-\-\lu  Zd 

•   fit  —  £ _____    •    r/ii  —   —  i 


We  also  have 


#> 

%y' 

*t~~ 


b  =  8    /     #*•  = 


FIG.  77. 


(175) 


in  which  A  is  the  area  of  the  triangle. 

If  the  base  is  parallel  to  the  neutral  axis,  as  in 
Fig.  77  a,  then,  by  similar  triangles, 
d  :  V) : :  f  d  —  y :  u 


TRANSVERSE   STRENGTH. 


167 


We  also  have 


(176) 


Equations  (173)  and  (175)  show  that  a  triangular  beam  which 
has  the  same  area  and  depth  as  a  rectangular  one,  is  only  half  as 
strong  as  the  rectangular  one.  JA£  &f~  *iw  /  /  7  £  I 

Some  authors  have  said  that  a  triangular  beam  is  twice  as 
strong  with  its  apex  up  as  with  it  down,  but  this  is  not  always 
the  case.  If  the  ultimate  resistance  of  the  material  is  the  same 
for  tension  as  for  compression,  the  beam  will  be  equally  strong 
with  the  apex  up  or  down. 

If  the  beam  is  made  of  cast  iron,  and  supported  at  its  ends,  it 
will  be  about  6  times  as  strong  with  the  apex  up  as  down ;  but 
if  the  beam  be  fixed  at  one  end,  and  loaded  at  the  free  end,  it 
will  be  about  6  times  as  strong  with  the  apex  down  as  with  it  up. 

141.    TRAPEZOIDAL  BEAM. — Required  the  strongest  trap- 
ezoidal beam  which  can  be  cut  from  a  given  triangular  one.\ 
c  Let  ABC  be  the  given  triangle, 

ABED  the  required  trapezoid, 
d  =  CG  =  the  longest  altitude, 


ft 

Fio.  78. 


z  —  CII  =  d,+  w,  and  v  =  DE. 

IJ  is  the  neutral  axis  of  the  trapezoid, 
which  passes  through  its  centre  of  gravity 
H.  We  may  then  find  :  — 


*  This  is  more  easily  solved  by  taking  the  moment  about  an  axis  through  the    ^f 
vertex  and  parallel  to  the  base,  and  using  the  formula  of  reduction.     See  Ap- 
pendix. 

f  See  an  article  by  the  author  in  ike,  Journal  of  Franklin  Institute,  vol.  xli., 
third  series,  p.  198. 

^    /  /  '  ~, .          "3      *..     ~ZT"       -  3 

U  ?        '        "?  *i    L/ 

*     "    '      '        '         <~v 

— —        --*  ^*_  .  y 


168  THE   RESISTANCE   OF   MATEEIALS. 

d      W-bv-v* 

^  =  *5X--5T^- 

d*  -5*  +  Pv  -  85V  +  85V  -  bv4  -  ^6 


j- 
r^FL 


T  tf  r55  +  J4^  -  85  V  4-  85V  -  ^  -  ^ 


which  is  to  be  a  maximum.  By  the  Calculus  we  find,  after  re- 
duction, that 

v3  -f  5fof  +  75^  -  J'  =  0, 

for  a  maximum,  which  solved  gives 

v  =  0.130935  or  0.135  nearly,  and  hence 
w  =  0.130936?  or  0.13(2     ....     (178) 

which  substituted  in  (177)  gives 

E^  =  0.545625^  .....    (179) 

&1  LJj 

Dividing  equation  (179)  by  equation  (176)  gives  1.09125  ;  hence 
from  (178)  and  (179)  we  infer  that  if  the  angle  of  the  prism  be 
taken  off  0.13  of  its  depth,  the  remaining  trapezoidal  beam 
will  be  1.091  times  as  strong  as  the  triangular  one,  which  is 
a  gain  of  over  9  per  cent. 

In  order  to  explain  this  paradox  it  must  be  granted  that  the 
condition  does  not  require  that  the  beam  shall  be  broken  in  two, 
but  that  a  fibre  shall  not  be  broken  —  in  other  words,  the  beam 
shall  not  be  fractured.  The  greatest  strain  is  at  the  edge,  where 
there  is  but  a  single  fibre  to  resist  it  ;  but,  after  a  small  portion 
of  the  edge  is  removed,  there  are  many  fibres  along  the  line 
DE,  each  of  which  will  sustain  an  equal  part  of  the  greatest 
strain. 

If  the  triangular  beam  were  loaded  so  as  to  just  commence 
fracturing  at  the  edge,  the  load  might  be  increased  9  per  cent. 
and  increase  the  fracture  to  only  thirteen-hundredths  of  the 
depth  ;  but  if  the  load  be  increased  10  per  cent,  it  will  break 
the  beam  in  two. 

These  results  are  independent  of  the  material  of  which  the 


TRANSVERSE    STRENGTH.  169 

beam  is  made.   If  the  beam  be  cut  off  £  the  depth,  its  strength 
is  found  from  equation  (177)  to  be 

Wd* 


0.465608 


12 


which  is  0.93101  of  equation  (176). 

Mr.  Couch  found,  for  the  mean  of  seven  experiments  on  tri- 
angular oak  beams  of  equal  length,  that  they  broke  with  306 
pounds.  The  mean  of  two  experiments  on  trapezoidal  oak 
beams,  made  from  triangular  beams  of  the  same  size  as  in  the 

O 

preceding  experiments,  by  cutting  off  the  edge  one-third  the 
depth  when  the  narrow  base  was  upward,  was  284.5  pounds. 
This  differs  by  less  than  half  a  pound  of  0.931  times  306 
pounds. 

149.  CYLINDRICAL  BEAMS.  —  The  moment  of  inertia  of  a 
circular  section  in  which  r  is  the  radius,  is 


/ 


.•.       =  iR«>-  (180) 

If  polar  co-ordinates  are  used,  we  have 

dudy  =  eded(j>, 

where  f  is  a  variable  radius  and  <£  a  variable  angle. 
Also  y  =  f  sin  $ 


/v     rz« 

=     I  I   ^Bin* 

*/0        I/O 


Fio.79. 

-  cos  2<£)  <fy  =  t*^  as  bef  ore-  \  $' 


For  a  circular  annulus  we  have 


^  '   n- 


170 


THE 


RESISTANCE   OF  MATERIALS. 


rl-W 


By  comparing  equations  (180)  and  (145)  we  see  that  the 
strength  of  a  cylindrical  beam  is  to  that  of  a  circumscribed 

rectangular  one  as  ^  :  -k  or  as£sj0$&  +  •  1. 

Oa  J  £~Q '}     '     j 

Also  the  strength  of  a  cylindrical  beam  is  to  that  of  a  square 
one  of  the  same  area  as  ^jf, Ad'  to  ^RAd  (df  being  the  diameter 
of  the  circle),  /) 

or  as  1  :  (%~^,  =  $•/*" J  or  as  1  : 1.18  nearly. 


143.     ELLIPTICAL   BKAIWS. 

Let  1)  =  the  conj  ugate  axis,  and 

d  =  the  transverse  axis  ;  then 
if  d  is  vertical  (Fig.  80),  we  have 
I  =  -giT  rid?  and  dl  =  ie?. 

If  5  is  vertical  (Fig.  81),  we  have 


FIG.  a. 


144.     PARABOLIC.  BEAMS. 


ft 

FIG.  82. 


FIG.  83. 


If  5  =  the  base,  and 

^  =  the  height  of  the  parabola,  and 
if  d  is  vertical  (Fig.  82),  we  have 

I  =  Tfy&VZ,  and  ^  = 


TRANSVERSE  STRENGTH.  171 

If  I  is  vertical  (Fig.  83),  then 

I  =  ^W,  and  d,  =  #>. 


145.     ACCORDING  TO  BARLOW'S  THEORY  WQ  have 

^ZPx  (181) 

which  must  be  integrated  between  the  proper  limits  to  include 
the  whole  section. 

If  the  neutral  axis  is  at  the  centre  of  the  sections,  and  the 
beam  is  rectangular,  we  have 

T 


Q 
which  reduced  gives 

•J-TW+ 

hence,  if  0  has  any  ratio  to  T,  the  law  of  resistance  in  solid  rec- 
tangular beams  is  the  same  as  for  the  common  theory  only, 

If  </>  =  T,  this  becomes 


172  THE   RESISTANCE   OF   MATERIALS. 


CHAPTER  VII. 

BEAMS  OF  UNIFORM  RESISTANCE. 

146.  GENERAL  EXPRESSION.  —  If  beams  are  so  formed 
that  they  are  equally  liable  to  break  at  every  transverse  section, 
they  are  beams  of  uniform  resistance,  and  are  generally  called 
Yearns  of  uniform  strength.  The  former  term  is  preferable,  be- 
cause it  applies  with  equal  force  to  all  strains  less  than  that  which 
will  produce  rupture.  In  such,  a  beam  the  strain  on  the  fibre 
most  remote  from  the  neutral  kxis  is  uniform  throughout  the 
whole  length  of  the  beam.  The  analytical  condition  of  such  a 
beam  is:  The  sum  of  the  moments  of  the  resisting  forces  must 
vary  directly  as  the  sum  of  the  moments  of  the  applied  forces  ; 
hence  equation  (171)  is  applicable  ;  or 


(182) 


which  must  be  true  for  all  values  of  x.     But  to  obtain  practical 
results  it  is  necessary  to  consider 

PARTICULAR   CASES. 
147.    BEAMS  FIXED  AT  ONE  END  AND  LOADED  AT  THE  FREE 

EIVD.  —  Required  the  form  of  a  beam  of  uniform  resistance 
when  it  is  fixed  at  one  end  and  loaded  at  the  free  end. 
1st.  Let  the  sections  be  rectangular,  and 

y  =  the  variable  depth,  and 
u  =  the  variable  width. 

Then  I  =  -^uy"  (see  equation  (51)), 
dl  =  %y,  and 

=  Px  =  the  variable  load.* 


*  For  2P#  use  the  general  moments  as  given  in  the  table  in  Article  101,  so 
far  as  they  are  applicable.  , 


BEAMS    OF    UNIFORM   RESISTANCE. 

Hence  equation  (182)  becomes 


173 


(183) 


a.  Let  the  breadth  be  constant ;    or  u  =  b ;  then  (183)  be- 
comes 

Paj  =  £Kfo/2, (184) 

which  is  the  equation  of  a  parabola,  whose  axis  is  horizontal 

6P 

and  parameter  is  y^r.     See  Fig.  84. 


FIG.  84. 


FIG.  85. 


b.  Suppose  the  depth  is  constant,  or  y  =  d.    Then  (183)  be- 
comes 

P«  =  |E^X     ........     (185) 

which  is  the  equation  of  a  straight  line  ;  hence  the  beam  is  a 
wedge,  as  in  Fig.  85. 

c.  If  the  sections  are  rectangular  and  similar,  then 

u  :  y  :  :  b  :  d 
b 


and  equation  (183)  becomes 


which  is  the  equation  of  a  cubical  parabola. 

2d.  Let  the  sections  be  circular.  Then 
I  =  -fairy*  (equation  (52),  in  which  y  is  the 
diameter  of  the  circle),  and  d^  =  %y  ;  hence 
(182)  becomes 


FIG.  86. 


which  is  also  the  equation  of  a  cubical  para- 
bola, as  shown  in  Fig.  86. 
3d.  Let  the  transverse  sections  be  rectangular,  and  I  con- 


174 


THE   RESISTANCE   OF   MATERIALS. 


stant,  the  breadth  and  depth  both  being  variable,  then  equation 
(182)  becomes 


(186) 


in  which  c  is  a  constant,  =  bd3,  b  and  d  being  the  breadth  and 
depth  at  the  fixed  end.  Equation  (186)  is  the  equation  of  the 
vertical  longitudinal  sections,  and  is  the  equation  of  an  hyperbola 


Fio.  88. 


referred  to  its  asymptotes.  See  Fig.  87.     If  the  value  of  y  from 
this  equation  be  substituted  in  the  equation  wf  =  c,  it  gives 

216FV 


which  is  the  equation  of  the  horizontal  longitudinal  sections  ; 
hence  they  are  cubical  parabolas,  as  in  Fig.  88.    For  x  and  u  =  0, 


y  =  oo,  and  for  x  =  Z,  u  =  b  =,>  73  *6 


..,, 

L  ^  4th.  If  the  breadth  is  the  w/th  power  of  the  depth,  and  the  sec- 
tions are  rectangular,  then  u  =  ^n,  and  equation  (183)  becomes 


which  is  the  general  equation  of  parabolas. 

148,       BEAMS     FIXED     AT     ONE    END    AND    UNIFORMLY 

.—  Required  the  form  of  a  learn  of  uniform  resistance 


BEAMS    OF   UNIFORM   RESISTANCE. 


175 


when  it  is  fixed  at  one  end  and  uniformly  loaded  over  its  whole 
length;  the  weight  of  the  beam  being  neglected. 

The  origin  of  co-ordinates  being  still  at  the  free  end,  we 
have 

wx  =  the  load  on  a  length  x,  and 
%wx*  r=  the  moment  of  the  load  (equation  (53)). 
Hence,  for  rectangular  sections,  equation  (182)  becomes 

%wxz  =  $B,uy*     .........          (188) 

a.  If  the  breadth  is  constant,  or  u  =  b  in  (188),  it  becomes 


which  is  the  equation  of  a  straight  line  ;  and  hence  the  beam 
will  be  a  wedge,  as  in  Fig.  89. 


TIG. 


b.  Let  the  depth  be  constant  ;  or  y  =  d  in  (188) 


a  parabola  whose  axis  is  perpendicular  to  the  axis  of  the  beam, 
as  in  Fig.  90. 

c.  Let  the  sections  be  similar  ;  — 


then  d  :  b  :  :  y  :  u  =  -&, 

.'.  equation  (188)  becomes 
a  semi-cubical  parabola,  as  in  Fig.  91. 


-,y*  ; 


FIG.  90. 


FIG.  91. 


176  THE   RESISTANCE   OF  MATERIALS. 

d.  Let  I  be  constant,  or  -fauy*  =  -f^bd3.  Then  equation  (182) 
becomes 

bd9 
^wx1  =  ^R  —  — ; — an  hyperbola  of  the  second  order. 

149.  PREVIOUS  CASES  COMBINED. — Required  the  form 
of  the  beam  of  uniform  resistance  when  it  is  fixed  at  one  end 
and  loaded  uniformly,  and  also  loaded  at  the  free  end. 

The  moment  of  applied  forces  is  P#+|-m£a;  hence  equation 
(182)  becomes,  for  rectangular  beams, 


Hence,  if  the  depth  is  constant,  P#  -f  \wy?  =  %Rud* ; — a 
parabola ; 

Hence,  if  the  breadth  is  constant,  P#  +  %wx*  =  %Rby* ; — an 
ellipse ; 

Hence,  if  the  sections  are  similar,  ~Px  -f  \wy?  —  ftl -^y* ; — a 

a      i 

semi-cubical  parabola. 

15O.     AVEIGHT  OF    THE    BEAM    CONSIDERED. Required 

the  form  of  the  beam  of  uniform  resistance  when  the  weight  of 
the  beam  is  the  only  load ;  the  beam  being  fixed  at  one  end  and 
free  at  the  other. 

a.  Let  the  sections  be  rectangular  and  the  breadth  constant. 

Let  x  =  AB ;  Fig.  92, 


b  —  the  breadth,  and 

S  =  the  weight  of  a  unit  of 

volume. 
Then    Cydx  —  the  area  of  ADC, 

and 
Sbjydx  =  the     weight     of 

ADC ;  PIO.  92. 

the  limits  of  the  integration 

being  0  and  x. 
If  F  is  the  centre  of  gravity  of  ADC ;  we  have,  from  the 

fxydx 
principles  or  mechanics,  the  distance  AJb  =  ~/»~77  * 


BEAMS   OF   UNIFOKM   EESISTANCE.  177 

The  moment  of  the  applied  forces  is  the  weight  of  ADC 

multiplied  by  the  distance  BF  =  x  —  AF.     Hence,  equation 
(182)  becomes 


which  reduced  gives 
2K 


which  is  the  equation  of  the  common  parabola,  the  axis  being 
vertical. 

QT> 

If  there  is  a  single  curve,  —^  is  its  parameter  ;  but  if  two 

T> 

curves,  as  in  the  figure,  -*  is  the  parameter  of  each. 
b.  Let  the  depth  be  constant.     In  a  similar  way  we  find 

This  solved  gives 


in  which  C  and  Cf  are  constants  of  integration,  and  involve 
the  position  of  the  origin  of  co-ordinates  and  direction  of  the 
curve  at  a  known  point. 

c.  Let  the  beam  be  a  conoid  of  revolution^  as  in  Fig.  93. 

We  have,  as  before 


FIG.  93. 


which  reduced  gives 

«?  =  if^y     ......     (191) 

2         y 


which  is  the  equation  of  the  common  parabola. 

d.  Suppose,  in  the  preceding  cases,  that  an  additional  load, 
P,  is  applied  at  the  free  end. 

Some  of  the  equations  which  result  from  this  condition  can- 
12 


178 


THE   RESISTANCE   OF   MATERIALS. 


not  be  integrated  in  finite  terms,  and  hence  the  curves  cannot 
be  classified. 

151.     BEAMS  SUPPORTED  AT  THEIR  ENBS. 

A.  Let  the  beam  ~be  supported  at  its  ends  and  loaded  at  the 
middle  point. 

For  this  case,  equation  (182)  becomes,  for  rectangular  sec- 
tions, 

JP«  =  iB«3f (192) 

a.  If  the  breadth  is  constant,  we  have 


which  is  the  equation  of  the  common  parabola. 

7) 


FIG.  94. 


FIG.  95. 


The  beam  consists  of  two  parabolas,  having  their  vertices, 
one  at  each  support,  as  in  Fig.  94. 
b.  If  the  depth  is  constant,  we  have 

JPar^KrfV;  -  (193) 

a  wedge,  as  in  Fig.  95. 

B.  If  the  learn  is  uniformly  loaded,  we  have  from  equations 
(74)  and  (182), 

\w  (Ix,  —  a;2)  =  %Ruy*  — if  rectangular,  and  if  the  breadth  is 

constant,  \w  (Ix  -  ar2)  =  -p%2 ; (194) 

an  ellipse,  Fig.  96. 

If  the  depth  is  constant,  %w  (Ix  —  a?2)  =  ^Rd\  a  parabola, 
Fig.  97. 


FIG.  96. 


FIG.  97. 


C.  Let  the  beam  have  an  uniform  load  and  also  an  uni- 
formly 'increasing  load  from  one  end  to  the  other  ^  as  in  Fig. 
98. 


BEAMS    OF   UNIFORM   RESISTANCE.  179 

Let  W  =  the  weight  of  the  uni-        y 

form  load, 

W,  =  the  weight  of  the  uni- 
formly        increasing 
load,  and 
V  =  the  reaction  of  the  sup- 
port at  the  end  which  has  the  least  load. 
Then   V=^W-{-^W1. 

Let  x  be  reckoned  from  A,  then  the  load  on  so  is 

W        W1/c3 

and  the  moment  of  this  reaction  and  load  on  a  section  which  is 
at  a  distance  x  from  A  is 

"W/v.2       TXT  ~,3 
QW+iWyc-^-Zg- (195) 

Zl>  6i 

which  equals  -J-R  by*  for  rectangular  beams  of  uniform  breadth. 
To  find  the  point  of  greatest  strain,  make  the  first  differential 
coefficient  of  (195),  equal  to  zero.  We  thus  find 


If  W  —  0,  this  gives 

x 

When  "W"  =  0,  this  becomes  the  case  of  water  pressing  against 
a  vertical  surface. 

152.  BEAMS  FIXED  AT  THEIR  ENDS. — If  the  beam  is 
fixed  at  its  ends  and  loaded  at  the  middle  with  a  weight,  P,  we 
have,  from  equations  (117)  and  (182),  when  the  breadth  is  uni- 
form, 

|P(Z-^)  =  iKfy2,  -    -,   (196) 

which  is  the  equation  of  a  parabola.     The  beam  really  consists 
of  four  double  parabolas  with  their 
vertices  tangent  to  each  other,  as  in 
Fig.  99.     The  vertices  are  %l  from 
the  end. 

If  the  load  were  uniform  we  would 
obtain,  in  a  similar  way,  a  beam  com-  FIG.  99. 

posed  of  four  wedges.     These  are  di- 
rect deductions  from  theory,  but  it  is  evident  that  there  is  some- 


ISO  THE   RESISTANCE   OF   MATERIALS. 

thing  wanting,  for  a  beam  like  Fig.  99  has  no  transverse 
strength.  The  same  result,  though  not  quite  so  glaringly  appa- 
rent at  first  sight,  exists  in  all  the  cases  which  we  have  discussed. 
For  instance,  in  figures  85,  86,  and  87  the  sections  at  the  free 
end  must  have  a  finite  value  to  resist  the  shearing  stress,  and 
the  beams  must  be  enlarged,  as  determined  in  the  next  article. 
If  the  section  is  reduced  to  naught,  it  can  sustain  no  weight.  In 
the  present  case,  there  is  neither  tension  nor  compression  at  A 
and  B,  as  was  shown  in  articles  99  and  100 ;  but  there  is  a 
transverse  shearing  stress  at  those  points,  and  there  must  be  suf- 
ficient transverse  section  to  resist  it.  The  same  remark  applies 
to  the  preceding  cases,  and  the  forms  must  all  be  modified  to 
meet  this  condition,  as  is  shown  in  the  next  article. 


153.    EFFECT    OF;  TRANSVERSE     SHEARING     STRESS     On 

modifying  the  forms  of  the  Yearns  of  uniform  resistance. 

The  value  of  the  transverse  shearing  stress  is  given  in  Article 
84.  For  instance,  in  the  case  of  a  beam  uniformly  loaded,  it  is 
V — wx  =  %wl  —wx  =  %w(l  —  2o?)  at  any  point  in  the  length. 
This  quantity,  divided  by  the  product  of  the  breadth  and  modu- 
les of  strength  for  transverse  shearing r,  gives  the  depth  neces- 
sary to  sustain  this  force.  Take,  for  example,  case  A,  Article 
134.  The  load  being  uniform,  we  have  Ss  =  ^w(l  —  2a?)  as 
given  above,  which  is  the  equation  of  a  straight  line,  Fig.  100, 
in  which 


FIG.  100.  FIG.  101. 

AB  =  \wl  ~  (&  x  modulus  of  shearing). 

Hence  we  would  at  first  thought  naturally  infer  that  the  form 
of  the  beam  of  uniform  strength  in  this  case  would  be  found  by 
adding  the  ordinates  of  the  straight  line,  AC,  to  the  correspond- 
ing ordinates  of  the  ellipse,  thus  giving  Fig.  101.  But  as  soon  as 
this  is  done  the  equation  of  moments  is  changed  ;  for  the  lever 
arm  of  the  force  is  increased,  and  the  moment  of  resisting 


BEAMS    OF    UNIFORM   RESISTANCE.  181 

forces  is  greater.  To  avoid  this  difficulty  we  would  add  the 
section  which  is  necessary  for  sustaining  the  shearing  stress, 
to  the  side  of  the  beam.  But  in  all  those  cases  where  the 
depth,  as  found  by  moments,  is  zero,  this  method  is  imprac- 
ticable, for  the  thickness  to  be  added  would  be  infinite.  It 
seems,  then,  that  to  solve  this  case  theoretically,  we  must  add 
some  arbitrary  quantity  to  the  depth  as  found  by  moments, 
which  quantity  shall  increase  the  section  so  as  to  fully  resist 
the  moment  of  the  applied  forces,  and,  in  addition  thereto, 
PARTLY  resist  the  shearing  stress,  and  then  a  section  must  be 
added  to  the  side  of  the  beam  which  shall  sustain  the  remain- 
der of  the  shearing  stress. 

Tabulated  values  of  shearing  stresses  for  several  of  the  cases 
which  have  been  considered.  The  values  in  the  fourth  column 
of  the  following  table  may  be  found  according  to  the  principles 
given  in  Article  84,  or  they  may  be  found  by  taking  the  first 
differential  coefficient  of  the  moments  of  applied  forces.* 


*  The  third  of  equations  (42a)  is  £P#  —  SF  sin.  a  x  x  =  EF,#,  and  since  the 
lever  arms,  x  and  y,  of  the  forces  are  always  linear  quantities,  we  may  enter 
under  the  sign  £  and  differentiate  them.  This  gives  EPdx  —  £F  sin  ackc  = 

dx 

(EFj)d^,  or    —  [EP  —  £F  sin  a]  =  £F,  which,  combined  with  the  second  of 
ay 

(42a),  gives  £P=EF  sin  a=8s.      Hence  we    have  this  simple  rule  :    Ss  is 
the  first  differential  coefficient  of  the  moments  of  the  applied  forces. 

"When  the  bending  moment  has  an  algebraic  maximum,  the  abscissa  of  the 
point  of  greatest  bending  stress  may  be  found  by  making  the  first  differential 
coefficient  of  the  moment  of  the  stress  equal  to  zero,  and  solving  for  x  ;  hence, 
in  tliu  case,  the  bending  moment  is  greatest  where  the  shearing  stress  is  zero. 


w 


182  THE    RESISTANCE    OF   MATERIALS. 

A  TABLE  OF  MOMENTS  AND   SHEARING   STRESS. 


Number  of 
the  Case.  > 
Seepagel||4. 

Condition  of  the  beam. 

General  moments  of  the 
Applied  Forces. 
See  page  130. 

Shearing  Stress. 

ss. 

I. 

Fixed  at  one  end  and 
P  at  the  free  end. 

P0.     Eq.  (35). 

P. 

II. 

Fixed  at  one  end  and 
uniformly  loaded. 

^wx*-.      Eq.  (40). 

wx. 

IV. 

Supported  at  the  ends 
and  P  at  the  mid- 
dle. 

iPz.     Eq.  (44). 

IP. 

Y- 

Supported  at  the  ends 
and  uniformly  load- 
ed. 

'  .  i  • 
iwlx-iwx\    Eq.  (48). 

\wl—  wx. 

VII. 

?ixed  at  one  end,  sup- 
ported at  the  other, 
and  P  anywhere. 

Tor  the  part  AD,  Fig. 
43,  Eq.  (64). 
For  the  part  DB,  Fig. 
43,  Eq.  (67). 

("ir«)p' 

VIII. 

Fixed  at  one  end,  sup- 
ported at  the  other, 
and  uniformly  load- 
ed. 

X4z2-3fe).  Eq.  (87). 

\w(8x—  31), 

.IX. 

Fixed  at  the  ends  anc 
P  at  the  middle. 

H«-4aj)P.       Eq.  (94). 

*. 

X. 

Fixed  at  the  ends  and 
uniformly  loaded. 

(102). 

%wl—  icx. 

Art.  150. 

Fixed  at  one  end,  anc 
the  weight  of  the 
beam  the  load. 

Sbxfydx—'bbJ'xydx. 

a$£dfc 

C.  Art.  151. 

Supported  at  the  ends 
uniform    load,   also 
load    uniformly   in 
creasing. 

Wy. 

*w+iw,---y 

BEAMS    OF   UNIFORM   RESISTANCE.  183 

If  a  beam  is  supported  at  its  ends,  and  loaded  with  several 

weights  P1?  P2,  P3,  etc.,  as  in 

p     P     P    p  Fig.  102,  we  may  readily  find 

the  shearing  stress  at  any  point 
I       I  by   article    84.      It    is    there 

shown  that  the  shearing  stress 


FlG  102  =  JfP,  where  J£P  equals  the 

algebraic  sum  of  all  the  ver- 

tical forces,  including  the  reaction  at  the  abutment.     Hence, 
we  have  for  the  shearing  stress 

between  the  end  and  Pj  =  Y  ; 

between  P1  and  P2  —  Y  —  PI  ; 

between  P2  and  P3  =  Y  -  P,  -  P2  ; 

between  P3  and  P4          =  Y  —  P,  —  P2  -  P3  ;  etc. 

If  the  weights  are  equal  to  each  other  =  P,  we  have  P  =  Pt 
=  P2  =P3,  etc.  ;  and  if  there  are  n  of  them,  and  they  are  sym- 
metrically placed  in  reference  to  the  centre  of  the  beam,  we  have 

Y  =  frP. 

If  n  is  even,  we  have,  at  the  centre  of  the  beam,  the 
transverse  shearing  stress  =  -J^P  —  %nP  =  0  -    -    (197)  ; 

and  if  n  is  odd,  there  will  be  a  weight  at  the  centre,  and  each 
side  of  the  central  weight  we  have 

transverse  shearing  stress  =  %nP  —  i(^±l)P  =  ±  -JP  -   (198).  , 


154.  UNSOLVED  PROBLEMS  —  Many  practical  problems  in  re- 
gard to  the  resistance  of  materials  cannot  be  solved  according 
to  any  known  laws  of  resistance.  Some  of  these  have  been 
solved  experimentally,  and  empirical  formulas  have  been  de- 
duced from  the  results  of  the  experiments,  Avhich  are  sufficiently 
exact  for  practical  purposes,  within  the  range  of  the  experi- 
ments. The  resistance  of  tubes  to  collapsing,  the  strength  of 
columns,  and  the  proper  thickness  of  the  vertical  web  of  rails, 
are  such  problems  which  have  been  solved  experimentally.  The 
following  problems  are  of  this  class,  and  have  not  been  solved. 
The  first  four  are  taken  from  the  Mathematical  Monthly  r,  Yol. 
L,  page  148. 

1.  Required  a  formula  for  the  strength  of  a  circular  flat  iron 


184 


THE   RESISTANCE    OF   MATERIALS. 


plate  of  uniform  thickness,  supported  throughout  its  circumfer- 
ence and  loaded  uniformly. 

2.  Required  the  strength  of  the  same  plate  if  the  edges  are 
bolted  down. 

3.  Required  the  equation  of  the  curve  for  each  of  the  pre- 
ceding cases,  that  they  may  have  the  greatest  strength  with  a 
given  amount  of  material. 

4.  In  the  preceding  problems,  suppose  that  the  plate  is  square. 

5.  Required  the  form  of  a  beam  of  uniform  strength  which  is 
supported  at  its  ends,  the  weight  of  the  beam  being  the  only 
load.     Suppose,  also,  that  it  is  loaded  at  the  middle. 

The  latter  part  of  this  problem  has  received  an  approximate 
solution  under  certain  conditions,  as  will  be  seen  from  the  fol- 
lowing experiments. 


155.  BEST  FORM  OF  CAST-IRON  BEAM  AS  FOUND  EXPERI- 
MENTALLY.— Cast-iron  beams  were  first  successfully  used  for 
building  purposes  by  Messrs.  Boulton  and  Watt.  The  form  of 
the  cross-section  of  the  beams  which  they 
used  is  shown  in  Fig.  103.  More  recent 
experiments  show  that  this  is  a  good 
form,  but  not  the  best. 

About  1822  Mr.  Tredgold  made  an 
experiment  upon  a  cast-iron  beam  of  the 
form  shown  in  Fig.  104,  to  determine 
its  deflection.  He  recommended  this 
form  for  beams. 

Mr.  Fairbairn  has  justly  the  credit  of 
making  the  first  series  of  experiments 
for  determining  the  best  fonn  of  the 
~beam.  These  experiments  were  prose- 
cuted by  himself  for  a  few  years,  beginning  about  1822,  and 
continued  still  later  by  Mr.  Hodgkinson. 

The   experiments   quickly  indicated  that  the  lower  flange 
should  be  considerably  the  largest. 


Fig.  103. 


Fig.  104. 


BEAMS   OF   UNIFORM   RESISTANCE. 


185 


The  following  experiments  were  made  by  Mr.  Hodgkinson 
(Fairbairn  on  Cast  and  Wrought  Iron,  p.  11). 


Fig.  105. 


105  shows  the  elevation  and  cross-section  of   a  beam 

o 

whose  dimensions  are  as  follows  :  — 

Area  of  top  rib  =1.75x0.42  =0.735  inches. 
Area  of  bottom  rib=1.77  x  0.39=0.690   " 


Thickness  of  vertical  rib,     -    - 
Depth  of  the  beam, 
Distance  between  the  supports, 
Area  of  the  whole  section,   -    - 
Weight  of  the  beam,    -    -    - 
Breaking  weight, 


0.29  " 
5.125  " 
54.00  " 

2.82  square  inches. 
-  36J  pounds. 
6,678   pounds. 


The  form  of  the  fracture  is  shown  at 
tension. 


n  r.    It  broke  by 


EXPERIMENT  IY. 


Dimensions.  Inches. 

Thickness  at  A  =  0.32 
«          "    B  =  0.44 
«         «    C  =  0.47 
«         "   FE  =  2.27 
«         «   DE  =  0.52 
Depth  of  the  beam  =  5.125 
Area  of  the  section  =  3.2  square  inches. 
Distance  between  the  supports  =  54  inches. 
Weight  of  casting  =  40^  Ibs. 
Deflection  with  5,758  Ibs.  =  0.25  inches. 

"  "    7,138   "    =0.37      " 

Breaking  weight  8,270,  Ibs. 


ioe. 


186  THE  RESISTANCE  OF  MATERIALS. 

EXPERIMENT  1£. 


FIG.  107. 

Dimensions  in  inches : — 

Area  of  top  rib  =  2.33  x  0.31  =  0.72. 
"      "  bottom  rib  =  6.67  x  0.66  =  4.4. 

Ratio  of  the  area  of  the  ribs  =  6  to  1. 

Thickness  of  vertical  part  =  0.266. 

Area  of  section,  6.4. 

Depth  of  beam,  5J-. 

Distance  between  the  supports,  54  inches. 

Weight  of  beam,  71  Ibs. 

This  beam  broke  by  compression  at  the  middle  of  the  length 
with  26,084  Ibs. 

It  is  probable  that  the  neutral  was  very  near  the  vertex  n,  or 
about  J  the  depth. 

EXPERIMENT  21. 


Bottom  Flanch 
Top  Planch. 


FIG.  108. 

This  was  an  elliptical  beam,  Eig.  108. 

Dimensions  in  inches  : — 

Area  of  top  rib  =  1.54  x  0.32  —  0.493. 

"     «  bottom  rib  =  6.50  x  0.51  =  3.315. 
Ratio  of  ribs,  6J  to  1. 
Thickness  of  vertical  part  =  0.34. 
Depth  of  beam,  5-J-. 
Area  of  the  section,  5.41. 
Distance  between  supports,  54  inches. 
Weight  of  beam,  70f  Ibs. 


BEAMS   OF   UNIFORM   RESISTANCE.  187 

Broke  at  the  middle  by  tension  with  21,009  Ibs. 

Form  of  fracture  b  n  r  ;  J  n  =  1.8  inches. 

As  these  beams  have  all  the  same  depth  and  rested  on  the 
same  supports,  4  feet  6  inches-  apart,  their  relative  strengths 
will  be  approximately  as  the  breaking  weight  r  divided  by  the 
area  of  the  cross  section. 

In  Experiment    1,  6,6T8   —  2.82  =  2,3 68  Ibs.  per  square  inch. 

«  «          14,8,270    -3.2    =2,584     «  « 

«  "          19,26,084-6.4    =4,075     «  « 

"  "          21, 21,009  —  5.41  =  3,883     "  « 

It  is  evident  from  these  experiments,  that  when  the  vertical 
rib  is  thin,  the  area  of  the  lower  rib  should  be  about  6  times 
that  of  the  upper.  In  the  19th  experiment  it  has  already  been 
observed  that  the  beam  broke  at  the  top,  and  in  the  21st  it 
broke  at  the  bottom,  although  the  lower  flange  was  larger  in 
proportion  to  the  upper  than  in  the  preceding  case,  and  the 
comparison  shows  that  they  were  about  equally  well  propor- 
tioned. They  should  be  so  proportioned  that  they  are  equally 
liable  to  break  at  the  top  and  bottom. 

A  beam  proportioned  so  as  to  be  similar  to  either  of  the  two 
last  forms  above  mentioned  may  be  called  a  "  TYPE  FORM." 

156.  HO»GKINSON»S  FORMULAS  for  the  strength  of  cast- 
iron  learns  of  the  TYPE  FORM. 

Let      TF=  the  breaking  weight  in  tons  (gross). 

a  =  the  area  of  bottom  rib  at  the  middle  of  the  beam. 

d  =  the  depth  of  the  beam  at  the  middle, 
and       I  =  the  distance  between  the  supports. 
Then  according  to  Mr.  Hodgkinson's  experiments  we  have 

TF=26-y-  when  the  beam  was  cast  with  the  bottom 

s  * 

rib  up,  and 

17=24^-  when  the  beam  is  cast  on  its  side. 

V 

157.  EXPERIMENTS  ON  T  RAILS. — Experiments  on  f  bars, 
supported  at  their  ends  and  loaded  at  the  middle,  gave  the  fol- 
lowing results : — * 

*  Mahan's  Civ.  Eng.,  pp.  88  and  89;  Barlow  on  the  Strength  of  Materials, 
p.  183. 


188  THE   RESISTANCE   OF  MATERIALS. 


Hot  blast  bar,  rib  upward,  JL  broke  with      -  -  1,120  pounds. 

"          "      "   downward,  T  broke  with  -        364       " 

Cold  blast  "      "   upward,  J.  broke  with.      -  -  2,352       " 

"         "      "   downward,  T  broke  with  -  -      980       " 

The  ratio  of  the  strengths  is  nearly  as  3  to  1,  but  according 
to  the  table  in  Article  47,  we  might  reasonably  expect  a  higher 
ratio.  If  a  greater  number  of  experiments  would  not  have 
given  a  higher  ratio,  we  would  account  for  the  discrepancy  by 
supposing  that  the  neutral  axis  moved  before  rupture  took 
place,  or  that  the  ratio  of  the  crushing  strength  and  tenacity  is 
less  for  comparatively  thin  castings  than  for  thick  ones.  It  is 
known  that  the  crushing  strength  of  thin  castings  is  proportion- 
ately stronger  than  thick  ones.  Hodgkinson  found  that  for 
castings  2,  2-J,  and  3  inches  thick,  the  crushing  strengths  were 
as  1  to  0.780  to  0.756  ;  and  Colonel  James  found  a  greater  in- 
crease— being  as  1  to  0.794  to  0.624.  See  also  Article  37. 

158.  WROUGHT-IRON  BEAMS.— The  treacherous  character 
of  cast-iron  beams,  on  account  of  the  internal  structure  of  the 
metal,  and  the  unseen  cracks  and  flaw's  which  may  exist,  has 
led  to  the  introduction  of  solid  wrought-iron  beams.  When  cast- 
iron  beams  were  first  used,  it  was  practically  impossible  to 
manufacture  solid  \vrought-iron  ones,  but  the  great  improve- 
ments which  have  been  made  since  then  in  the  processes  of 
manufacturing,  have  not  only  niade  their  construction  possible, 
but  they  have  enabled  the  manufacturer  to  produce  them  so 
cheaply  as  to  bring  them  within  the  means  of  those  who  desire 
such  articles.  At  Trenton  and  Pittsburg  they  make  rolled 
beams  from  a  single  pile,*  but  it  is  stated  that  by  this  method 
they  can  make  beams  only  about  nine  inches  in  depth.  At 
Buffalo  and  Phoenixville  they  use  Mr.  John  Griffin's  patent, 
which  consists  in  rolling  the  flanges  separately,  piling  the  plates 
for  the  web  between  them,  and  then  rolling  and  welding  the 
whole  together.  By  this  method  they  can  make  beams  at  least 
twenty  inches  deep,  and  of  any  desired  length.  There  is  no 
attempt  to  make  them  of  uniform  strength.  They  are  of  the 
double  T  (I)  pattern,  and  of  uniform  section  throughout. 

*  Jour,  Frank.  Inst.,  Vol.  86,  p.  331. 


TOKSION.  1S9 


' 

CHAPTEK  VIII. 

TORSION. 

159*  TORSIVE  STRAINS  are  very  common  in  machinery. 
In  all  cases  where  a  force  is  applied  at  one  point  of  a  shaft  to 
turn  (or  twist)  it,  and  there  is  a  resisting  force  at  some  other 
point,  the  shaft  is  subjected  to  a  torsi ve  strain.  The  wheel  and 
axle  is  a  familiar  case  in  which  the  axle  is  subjected  to  this 
strain.  To  produce  torsion  without  bending,  two  equal  and  par- 
allel forces,  acting  in  opposite  directions,  and  lying  in  a  plane 
which  is  perpendicular  to  the  axis  of  the  piece,  must  be  so  applied 
to  the  section  that  the  arms  of  the  forces  shall  be  equal.  In  other 
words,  mechanically  speaking,  a  Couple  whose  axis  coincides 
with  the  axis  of  the  piece,  must  be  applied  to  the  piece.  If 
only  a  single  force,  P,  is  applied,  as  in  Fig.  109,  the  piece  is 
pushed  sidewise  at  the  same  time  that  it  is  twisted ;  but  the 
amount  of  twisting  is  the  same  as  if  the  force,  P,  were  divided 
into  two,  each  equal  £P,  and  each  of  these  acted  on  opposite 
sides  of  the  axis  and  in  opposite  directions,  and  at  a  distance 
from  the  axis  equal  AB,  Fig.  109.  For,  the  moment  of  the 
couple  thus  formed,  is  £Px2x AB^P.AB,  which  is  the 
moment  of  P. 

16O.  THE  ANGLE  OP  TORSION  is  the  angle  through  which 
a  fibre  whose  length  is  unity,  and  which  is  situated  at  a  unit's 
distance  from  the  axis,  is  turned  by  the  twisting  force.  It 
depends  for  its  value,  in  any  case,  upon  the  elastic  resistance  to 
torsion,  as  wrell  as  upon  the  dimensions  of  the  piece  and  the 
twisting  force.  The  analysis  by  which  its  value  is  determined 
is  founded  upon  the  following  hypotheses,  which  are  approxi- 
mately correct. 

'/U*r^    7^ 

/  v 


190 


THE   RESISTANCE   OF   MATERIALS. 


1st.  The  resistance  of  any  fibre  to  torsion  varies  directly  as  its 
distance  from  the  axis  of  the  piece. 

2d.  The  angular  amount  of  torsion  of  any  fibre  between  any 
two  sections,  or  the  total  angle  of  torsion,  varies  directly  as  the 
distance  between  them. 

It  is  found  by  experiment,  that  these  hypotheses  are  suffi- 
ciently exact  for  cylinders  and  regular  polygonal  prisms  of 
many  sides.  They  assume  that  transverse  sections  which  were 
plane  before  twisting,  remain  so  while  the  piece  is  twisted,  but 
in  reality  the  fibres  which  were  parallel  to  the  axis  before 
being  twisted  are  changed  to  helices,  and  this  operation  pro- 
duces a  longitudinal  strain  upon  the  fibres ;  and  this,  in  turn, 
changes  the  transverse  sections  into  warped  surfaces.* 

To  find  the  angle  of  torsion  :— 
Let  I  —  AD  =  the  length  of  the  piece, 

Fig.  109. 

a  =  AB  —  the  lever  arm  of  P. 
P  =  the  twisting  force. 
a,  •=.  a  A3)  =  the  total  angle  of  tor- 
sion,   or    angle    through 
which  Aa  has  been  twist- 
ed. 

0  -  j  =  "  The  Angle  of  Torsion," 

*  Fig.  109. 

— supposed  to  be   small. 

f(Q,  <P)  =  the  equation  of  a  transverse  section,  and 
G  =  the  coefficient  of  the  elastic  resistance  to  torsion,  which 
is  the  force  necessary  to  turn  one  end  of  a  unit  of  area  and 
unit  of  length  of  fibres  through  an  angle  unity,  the  vertex 
of  the  angle  being  on  the  axis  of  torsion,  one  end  of  the  fibres 
being  fixed  and  the  twisting  force  being  applied  directly  to 
the  other  end,  and  acting  in  the  direction  of  a  tangent  to  the 
arc  of  the  path  described  by  the  free  end. 

As  a  unit  of  fibres  cannot  be  placed  so  that  all  of  them  will 
be  at  a  unit's  distance  from  the  axis,  we  must  suppose  that  the 
resistance  of  a  very  thin  annulus,  which  is  at  a  unit's  distance, 
is  proportional  to  that  of  a  unit  of  section  •  or  the  resistance 


*  Resume  des  Lecons,  Navier,  Paris,  1864,  p.  276  and  several  other  pages 
following. 


TORSION.  191 

of  an  element  at  a  units'  distance  from  the  axis  is  G  multiplied 
by  its  area ;  which  expressed  analytically  is 

Gpdpd<p, 
and  according  to  the  first  law 

Gp2dpd(j>  =  the    resistance    of    any 

fibre  whose  length  is  unity,  to  being  twisted  through  an  angle 
unity  ;  and  the  moment  of  resistance  —  Gp*dpd<f>  for  an  angle 
unity ;  and  for  any  angle  6  the  moment  is,  according  to  the 
second  law, 

G6p5dpd<l> 

and  the  total  moment  equals  the  moment  of  the  applied  force, 
or  moments  of  the  applied  forces  ;  hence 


where  Ip  is  the  polar  moment  of  inertia  of  the  section. 

/r       /*2?r 
/    padpd(f>  =  ^7T/>4  (199) 
_  i/O 


7T0/-4  7T*/'4 

or,  6  — 


161.  THE  VALUE  OF  THE  COEFFICIENT  G  may  be  found 
from  equation  (200).  M.  Cauchy  found  analytically  on  the  con- 
dition that  the  elasticity  of  the  material  was  the  same  in  all 
directions,  that  G  =  f  E.*  M.  Duleau  found  experimentally 
that  G  is  less  than  f  E,  and  nearly  equal  |-  E,  f  and  M.  Wert- 
heim  found  G  =  f  E  nearly.  £  M.  Duleau's  experiments  gave 
the  following  mean  values  for  G :  f 

Pounds. 

Soft  iron 8,533,680 

Iron  bars      -    - 9,480,917 

English  steel 8,533,680 

Forged  steel  (very  fine) 14,222,800 

Cast  iron    -    -    -    - 2,845,600 

*  Resume  des  Lemons,  Navier,  Paris,  1856,  p.  197. 
f  Resistance  des  Materiaux,  Morin,  p.  461. 
t  L'Engineer,  1858,  p.  52. 


192 


THE   RESISTANCE   OF   MATERIALS. 


Copper 
-Bronze 
Oak  - 
Pine  - 


Pounds. 

6,209,670 

1,516,150 

568,912 

615,472 


Example. — If  an  iron  shaft  whose  length  is  5  feet,  and  diameter  2  inches, 
is  twisted  through  an  angle  of  7  degrees  by  a  force  P  =  5,000  Ibs.,  acting  on  a 
lever,  a  =  6  inches,  required  G-.  The  7  degrees  is  first  reduced  to  arc  by  multi- 
plying it  by  ~ -,  which  gives  a  =  -^-,  and  Eq.  (200)  gives, 


180 


2  x  5000 


60 


^  9,697,000  Ibs. 


(3.1416)3  x 

Y-  =i 

TORSION  ^ENDUIjTJOT. — If  a  prism  is  suspended  from  its 
upper  end,  and  supports  an  arm  at  its  lower  end,  and  two  weights  each  equal 
•£  W  are  fixed  on  the  arm  at  equal  distances  from  the  prism,  and  the  prism  be 
twisted  and  then  left  free  to  move,  the  torsional  force  will  cause  an  angular 
movement  of  the  arm  until  the  fibres  are  brought  to  their  normal  position, 
after  which  they  will  be  carried  forward  into  a  new  position  by  the  inertia 
of  the  moving  mass  in  the  weights  ^  W  until  the  torsional  resistance  of  the 
prism  brings  them  to  rest,  after  which  they  will  reverse  their  movement,  and 
an  oscillation  will  result.  The  conditions  of  the  oscillation  may  easily  be  inves- 
tigated if  the  prism  is  so  small  that  its  mass  may  be  neglected. 

For,  equation  (200)  readily  gives  : 


from  which  it  appears  that  the  torsional  force  P  varies  as  the  space   (aa)   over 
which  it  moves. 

It  is  a  principle  of  mechanics  that  the  moving  force  varies  directly  as  the 
product  of  the  moving  mass  multiplied  by  the  acceleration.  Hence,  if  x  =  (&a), 
the  variable  space,  and  t  =  the  variable  time,  and  M  =  the  mass  moved,  and 
observing  that  t  and  x  are  inverse  functions  of  each  other,  and  the  above  prin- 
ciple of  mechanics  gives  the  following  equation :  — 

Ms  *Gr* 


Multiplying  both  members  by  the  dx,  gives 
W    dxd*x  *G 


where  W  is  the  weight  of  the  mass  moved,  and  g  is  the  acceleration  due  to 
gravity.     The  oscillations  commence  at  the  extremity  of  an  arc  whose  length 


TOK8IOX.  193 

is  s,  at  which  point  the  velocity  is  zero.     The  integral  of  the  last  equation 
between  the  limits  s  and  x  is 


(*'- 

Ut/  «TYW*" 

A  second  integral  gives 
t 


~\p%?       [Sin 
which  is  the  time  of  half  an  oscillation.     For  a  whole  oscillation  : 


This  is  essentially  the  theory  of  Coulomb's  torsion  pendulum.  A  torsion 
pendulum  was  used  by  Cavendish  in  1778  to  determine  the  density  of  the 
earth.  (See  Royal  Philosophical  Transactions:  London,  Vol.  18,  p.  388.)  He 
found  the  mean  density  of  the  earth  by  this  method  to  be  5.48  times  that  of 
water.  This  is  considered  the  most  reliable  of  all  the  known  methods,  but 
the  results  of  other  methods  exceed  the  value  given  above  by  a  small  amount 
only,  thereby  confirming  this  result  and  showing  that  the  mean  density  of  the 
earth  is  about  5-J  times  that  of  water. 

1 63.  RUPTURE  BY  TORSION — The  resistance  which  .a  bar 
offers  to  a  twisting  force  is  a  torsional  shearing  resistance,  and 
in  regard  to  rupture,  the  equation  of  equilibrium  is  founded 
upon  the  following  principles  : — 

1st.  The  strain  upon  any  fibre  varies  directly  as  its  distance 
from  the  axis  of  torsion ;  and, 

2d.  The  sum  of  the  moments  of  resistances  of  the  fibres 
equals  the  sum  of  the  moments  of  the  twisting  forces. 

Let  S  =  the  MODULUS  OF  TORSION,  that  is,  the  ultimate  resist- 
ance to  torsion  of  a  unit  of  the  transverse  section  which  is 
most  remote  from  the  axis  of  torsion.  It  is  the  ultimate  shear- 
ing resistance  to  torsion,  but  may  be  used  for  any  shearing  strain 
which  is  less  than  the  ultimate, 

dl  =  the  distance  of  the  most  remote  fibre  from  the  axis  of 
torsion, 

f  (e,  0)  =  the  equation  of  the  section, 

P  =  the  twisting  force,  and 

a  =  the  lever  arm  of  P. 

Ip  =z  the  polar  moment  of  inertia  of  a  section. 

Then  pdpdQ  =  dA.  =  the  area  of  an  element  of  the  section  ; 
•  13 


194  THE    RESISTANCE    OF   MATERIALS. 

Spdpd(j>  =  the  shearing  strain  of  the  most  re- 
mote element  ;  and,  by  the  first  prin- 
o  ciple  given  above, 

—pdpdcj)  =  the  shearing  strain  of    any  element, 
1  which  is  at  a  unit's  distance  from  the 

axis  of  torsion,  according  to  the  first 
principle  above  ;  and  from  the  same 
principle  we  have 

_  p*dpd<j>  =  the  shearing  strain  of   any  element, 
d\  ,    -     and  this,  multiplied  by  the  distance  of 

v 


the  element,  p,  from  the  axis,  gives 
ent  of  resistance  to  torsion 
Hence,  according  to  the  second  principle  we  have 


S 

—  p3dpd<f>  =  the  moment  of  resistance  to  torsion. 


=    *  -  (201)     • 

For  circular  sections,  we  have  already  found,  Eq.  (199), 

Ip  =  *  Trr*. 

For  square  sections,  whose  sides  are  5,  we  may  find  *  A~ 

4 


=  -J-&4,  and  A  = 


164.  PRACTICAL  FORMULAS.—  Equations  (199)  and  (201) 

give  for  cylindrical  pieces,  observing  that  di  =  r,       7  ft. 

......     -     (202) 


If  cylindrical  pieces  are  twisted  off  by  forces  which  form  a 
couple,  and  P,  &,  and  r  measured,  the  value  of  S  may  be  found 
from  equation  (202).  Cauchy  found  S  =  |-  Tt,f  which  is  con- 
sidered sufficiently  ex^et-  when  a  proper  coefficient  of  safety  is 
used.  Calling  S  =  25,000  pounds  for  iron,  and  using  about  a 


*  We  have  /"/TOA  ==  /TiBa+yA  dA  =  Cxn-dk+ -Jy'dA,  that  is,  the  polar 

moment  equals  the  sum  of  the  rectangular  moments,  the  origin  being  the  same 
in  both  cases.     In  this  case  the  origin  being  at  the  centre  of  the  square,  we 

have  y  a;2dA  ^Cy^dK  .'.Ip  =  2fadT=?k  x  &#  (see  Eq."(^).   £" j  j 


f  Resume  des  Legons,  Navier.     Paris,  1856,    pp.  193-203,  and  p.  507. 

To 


•V-  / 

?r~  —  < 
D3 

TORSION. 

five-fold  security ;  and  S  —  8,000  pounds  for  wood,  and  using 
about  a  ten-fold  security,  and  we  may  use  for 

ROUND  IRON  SHAFTS  (wrought        /r\ 

or  cast),  diameter 
SQUARE  IRON  SHAFTS  (wrought         : 

or  cast),  side  of  the  squarejt==  -^y  ra  I 
SQUARE  WOODEN  SHAFTS,  _  I 

side  of  the  squares         *v  =•  •J-  ^P#/ 

The  dimensions  given  by  these  formulas  are  unnecessarily 
large  for  a  steady  strain,  but  shafts  are  frequently  subjected  to 
sudden  strains,  amounting  sometimes  to  a  shock,  and  in  these 
cases  the  results  are  none  too  large. 

Practical  formulas  may  also  be  established  on  the  condition 
that  the  total  angle  of  torsion  shall  not  exceed  a  certain 
amount.  Making  G  =  f  E,  and  solving  (200)  in  reference  to  r, 
and  we  have  for  cylindrical  shafts, 

&  ^   ^     ^    ~      **-£ 

;>£^       jf^Tt^ 
and  similarly  for  square  shafts, 


_    4/l6P^ 

-VT 


In  these  expressions  P  should  not  be  so  great  as  to  impair 
the  elasticity,  —  say  for  a  steady  strain  P  should  not  exceed  the 
values  given  by  equation  (203). 

If  x°  is  given  in  degrees,  it  is  reduced  to  arc  by  multiplying 

it  by  -  -  so  that  *  —  -  —  -«°  ;  hence  the  preceding  equations  be- 
loU  loO 

come  :  for  cylindrical  iron  shafts, 


and  for  square  iron  shafts, 


Examples—  1.  A  round  iron  shaft  fifteen   feet  long,  is  acted  upon  by  a, 
weight  P  =  2,000  Ibs.  applied  at  the  circumference  of  a  wheel  which  is  on  the 


196  THE   RESISTANCE    OF   MATERIALS. 

shaft,  the  diameter  of  the  wheel  being  two  feet ;  what  must  be  the  diameter 
of  the  shaft  so  that  the  total  angle  of  torsion  shall  be  2  degrees  ? 
If  the  shaft  is  cast-iron  E  =  16,000,000,  and 


-.**«# 


2000  x  12  x  15  x  12 
gxlMOO|000 


2.  A  round  wooden  shaft,  whose  length  is  8  feet,  is  attached  to  a  wheel 
whose  diameter  is  8  feet.  A  force  of  200  pounds  is  applied  at  the  circumfer- 
ence of  the  wheel,  what  must  be  the  diameter  of  the  shaft  so  that  the  total 
angle  of  torsion  shall  not  exceed  2  degrees  ? 


12 

J    =4.35  inches. 


For  further  information  upon  this  subject  see  "Resistance  des  Materiaux," 
Navier  ;  Paris,  1856,  pp.  237-509,  and  the  exhaustive  articles  of  Chevandier 
and  Wertheim  in  "  Annales  des  Chemie  et  Physique,"  Vol.  XL.  and  Vol.  L. 

1  65.    RESULTS  OF   WERTHEIIJI'S  EXPERIMENTS.  -  A  f  6W 

years  since  M.  Gr.  Wertheim  presented  to  the  French  Academie 
des  Sciences  an  exhaustive  paper  upon  the  subject  of  torsion,  the 
substance  of  which  was  published  in  the  Annales  de  Chimie  et 
de  Physique,  Yol.  XXIII.,  1st  Series,  and  Yol.  L.,  3d  Series. 
These  articles  would  make  a  volume  by  themselves,  and  hence 
we  will  content  ourselves  at  this  time  with  presenting  his 

CONCLUSIONS. 

When  a  body  of  three  dimensions  is  subject  to  torsion  the 
following  facts  are  observed  :  — 

1st.  The  torsion  angle  will  consist  of  two  parts,  one  tempo- 
rary, the  other  permanent  ;  the  latter  augments  continually, 
though  not  regularly. 

2d.  The  temporary  displacements  augment  more  and  more 
rapidly  than  the  moments  of  the  applied  couple's,  and  the  increase 
of  the  mean  angle,  which  in  hard  bodies  continues  until  rup- 
ture, in  soft  bodies  continues  only  to  the  point  where  the  body 
commences  to  suffer  rapid  and  continuous  deformation. 

-3d.  The  temporary  angles  are  not  rigorously  proportional  to 
the  length,  and,  all  else  being  equal,  the  disproportionality  in- 
creases in  measure  as  the  bar  becomes  shorter. 

4th.  In  all  homogeneous  bodies,  torsion  caused  a  diminution 

O  ' 

of  the  volume,  which  is  proportional  to  the  length  and  square 
•of  the  angle  of  torsion,  and  each  point  of  the  body,  instead  of 


TOKSION.  1 97 

describing  an  arc  of  a  circle,  follows  the  arc  of  a  spiral.  The 
condensation  of  the  body  increases  from  the  centre  to  the  cir- 
cumference. 

5th.  In  bodies  with  three  angles  of  elasticity,  the  change  of 
volume  and  resistance  to  torsion  are  functions  of  the  three  axes, 
and  the  relation  between  them  may  be  such  that  the  volume 
will  augment. 

6th.  Circular  or  turning  vibrations  of  great  amplitude  are 
difficult  to  produce,  and  as  small  angles  of  torsion  only  are  used, 
the  preceding  conclusions  apply  to  this  case. 

7th.  Rupture  produced  by  torsion  usually  takes  place  at  the 
middle  of  the  length  of  the  prism ;  it  commences  at  the  dan- 
gerous points,  and  operates  by  slipping  in  hard  bodies  and  by 
elongation  in  soft  ones. 

8th.  With  regard  to  the  influence  of  the  figure  and  absolute 
dimensions  of  the  transverse  sections  of  the  bodies,  we  derive 
the  following  conclusions : — 

9th.  In  homogeneous  circular  cylinders  the  diminution  of  the 
volume  is  equal  to  the  original  volume  multiplied  by  the  prod- 
uct of  the  square  of  the  radius,  and  the  angle  of  torsion  for  a 
unit  of  length  (the  angle  being  always  very  small).  Further, 
under  torsion  the  radius  of  the  cylinder  equals  the  primitive 
radius  multiplied  by  the  sine  of  the  angle  of  inclination  of  the 
helicoidal  fibres.  This  last  gives  a  means  of  calculating  the 
diminution  of  volume.  But  in  reality  the  twisted  cylinder 
takes  the  form  of  two  frustra  of  cones  joined  at  the  smaller 
bases ;  and  although  this  does  not  sensibly  affect  the  theoretical 
results  for  long  cylinders,  yet  it  deprives  our  formulas  of  all 
their  value  in  ordinary  practical  cases. 


198  THE   RESISTANCE    OF    MATERIALS. 


CIIAPTEE  IX. 

EFFECT  OF  LONG-CONTINUED  STRAINS— OF  OFT-REPEATED 
STRAINS,  AND  OF  SHOCKS— REMARKS  UPON  THE  CRYSTAL- 
LIZATION OF  IRON. 

EFFECT  OF  LONG-CONTINUED    STRAINS.     , 

1 6O.  GENERAL  EFFECT. — The  values  of  the  coefficients  of 
elasticity  and  the  modnlii  of  tenacity,  crushing,  and  of  rupture 
were  determined  from  strains  which  were  continued  for  a  short 
time — generally  only  a  few  minutes — or  until  equilibrium  was 
apparently  established ;  and  yet  it  is  well  known  that  if  the 
strain  is  severe,  the  distorsion,  whether  for  extension,  compres- 
sion, or  bending,  will  increase  for  a  long  time  ;  and  as  for  rup- 
ture, it  always  takes  time  to  break  a  piece,  however  suddenly 
rupture  may  be  produced.  By  sudden  rupture  we  only  mean 
that  it  is  produced  in  a  very  short  time. 

The  increased  elongation  due  to  a  prolonged  duration  of  the 
strain  beyond  a  few  minutes,  will  affect  the  coefficient  of  elas- 
ticity but  very  slightly,  for  the  strains  which  are  used  in  deter- 
mining it  are  always  comparatively  small,  and  the  greater  part 
of  the  effect  is  produced  immediately  after  the  stress  is  applied. 
Still,  if  the  distortion  should  go  on  indefinitely,  no  matter  how 
slowly,  the  elasticity,  and  hence  the  coefficient,  would  be  greatly 
modified  by  a  very  great  duration  of  the  stress,  however  small 
the  stress  may  be ;  and  at  last  rupture  would  take  place.  If  the 
basis  of  this  reasoning  be  well  founded,  we  might  reasonably 
fear  the  ultimate  stability  of  all  structures,  and  especially  those 
in  which  there  are  members  subjected  to  tension.  But  the  con- 
tinued stability  of  structures  which  have  stood  for  centuries, 
teaches  us,  practically  at  least,  that  in  all  cases  in  which  the 
strain  is  not  too  severe,  equilibrium  is  established  in  a  short 
time  between  the  stresses  and  strains,  and  in  such  cases  the 
piece  will  sustain  the  stress  for  an  indefinitely  long  time. 


TORSION. 


199 


.  HOI>GKINSON>S  EXPERIMENTS. — The  results  of  the 
experiments  which  are  recorded  in  Article  XL.,  page  48,  show 
that  in  one  case  the  compression  increased  with  the  duration  of 
the  strain  for  three-fourths  of  an  hour.  In  the  case  of  exten- 
sion on  another  bar,  as  shown  in  Article  YIL,  page  7,  it  ap- 
pears that  the  same  weight  produced  an  increased  elongation 
for  nine  hours ;  but  during  the  last,  or  tenth  hour,  there  was  no 
increase  over  that  at  the  end  of  the  ninth  hour. 

In  both  these  cases  the  strain  was  more  than  one-half  that  of 
the  ultimate  strength. 

168.  VICAT'S  EXPERIMENTS. — M.  Yicat  took  wrought- 
iron  wire  an<\  subjected  it  to  an  uniform  stress  for  thirty-three 
months.  The  elongations  produced  by  the  several  weights  were 
measured  soon  after  the  weights  were  applied,  and  total  lengths 
determined  from  time  to  time  -during  the  thirty-three  months. 
It  was  found  for  all  but  the  first  wire,  as  given  in  the  following 
table,  that  the  increased  elongations  after  the  first  one  were 
very  nearly  proportional  to  the  duration  of  the  stress.  (Annales 
de  Chemie  et  Physique,  Vol.  54,  2d  series.) 

TABLE 

Of  the  Result*  of  M.    VicaVs  Experiments  on  Wrought-iron  Wire. 


Amount  of  Strain. 

QJ     O     QJ 
'     *l 

8  %  A 

Sal 

*  '3  8    . 

if  If 

Increased  Elongation  after  33 
months. 

i  of  its  ultimate  tensile  strain.  .  . 
£  of  its  ultimate  tensile  strain.  .. 

No  additional  increase. 
0.027  of  an  inch  per  foot. 

£  of  its  ultimate  tensile  strain.  .. 
£  of  its  ultimate  tensile  strain  .  .  . 

lit! 
HJ 

0.040  of  an  inch  per  foot. 
0.061  of  an  inch  per  foot. 

169,  FAIRBAIRN'S  EXPERIMENTS. — Fairbairn  made  ex- 
periments upon  several  bars  of  iron,  which  were  subjected  to  a 
transverse  strain,  the  results  of  some  of  which  are  recorded  in 
the  following  tables.  (See  Cast  and  Wrought-iron,  by  Wm. 
Fairbairn.)  The  bars  were  four  feet  six  inches  between  the 
supports,  and  weights  were  applied  at  the  middle,  and  permitted 


200 


THE   RESISTANCE    OF   MATERIALS. 


to  remain  there  several  years,  as  indicated  by  the  tables.  The 
deflections  were  noted  from  time  to  time,  and  the  results  were 
recorded. 

TABLE  I. 

In  which  the  Weight  Applied  was  336  pounds. 


i 

i 

; 

» 

1 

l| 

71 

l| 

TEMPERATURE. 

^£  -rH 

«w          • 

o    . 

3-S 

J.s 

°  C^ 

•°  ft 

rt 

•^    *c 

$1 

T3  .2 

5  2 

.2  g  w 

I* 

° 

w 

K 

March  11,1837.... 

1.270 

1.461 

Cold-blast, 

78° 

June  3,  1838 

1  316 

1  538 

0  661  •  1 

72° 

July  5,  1839 

1  305 

1  533 

Hot-blast 

61° 

June  6,  1840  

1  303 

1  520 

0  694  •  1 

50° 

November  22,  1841. 

1.306 

1.620 

58° 

April  19   1842 

1  308 

1  620 

Mean  

1.301 

1.548 

Previous  to  taking  the  observations  in  November  and  April 
the  hot-blast  bar  had  been  disturbed. 

In  regard  to  this  experiment  Mr.  Fairbairn  remarks  : — "  The 
above  experiments  show  a  progressive  increase  in  the  deflec- 
tions of  the  cold-blast  bar  during  a  period  of  five  years  of  0.031 
of  an  inch,  and  of  0.087  of  the  hot-blast  bar."  The  numerical 
results  are  found  by  comparing  the  first  deflection  with  the 
mean  of  all  the  observed  deflections.  But  an  examination  of 
the  table  shows  that  the  greatest  deflection,  which  was  ob- 
served in  both  cases,  was  at  the  second  observation,  which  was 
about  a  year  and  a  quarter  after  the  weight  was  applied,  and 
during  the  next  two  years  the  deflections  decreased  0.015  of  an 
inch  for  the  cold-blast,  and  0.018  of  an  inch  for  the  hot-blast 
bar.  After  this  the  deflections  appear  to  increase  for  the  cold- 
blast  bar  0.005  of  an  inch  the  next  two  years.  Considering  all 
the  particulars  of  these  experiments  it  does  not  seem  safe  to 
conclude  that  the  deflections  would  go  on  increasing  indefinitely 


TOKSIOX. 


201 


with  a  continuance  of  the  load.      Admitting  that  the  small  in 
crease  of  deflections  during  the  last  two  years  are  correct  and 
not  due  to  errors  of  observation,  and  we  see  no  reason  why  the 
deflections  would  not  be  as  likely  to  decrease  after  a  time  as 
they  were  after  the  first  year. 

TABLE  II. 

In  wJiich  the  Bar  was  Loaded  with  392  pminds. 


TEMPERATURE. 

o 

ll 

Cold-blast,—  deflec- 
tion in  inches. 

Hotblast,—  deflec- 
tion in  inches. 

5! 
iff 

March  6,  1837  

1.684 

1  715 

78' 

June  23   1838 

1  824 

1  803 

For  cold-blast 

72° 

July  5  1839 

1  824 

1.798 

0  771  •  1 

61° 

June  6,  1840  

1  825 

1  798 

For  hot-blast 

50° 

58° 

November  22,  1841. 
April  19,  1842  

1.829 
1.828 

1.804 

1.812 

0.805  :  1 

Mean             

1.802 

1  788 

Here  we  see  a  general  increase  in  the  deflections  from  year 
to  year,  being  very  regular  in  the  cold  blast,  and  quite  irregular 
in  the  hot  blast.  But  we  observe  that  the  increase  is  exceed- 
ingly small  after  the  first  year,  being  only  0.004  of  an  inch  in 
the  cold  blast  bar,  and  0.009  of  an  inch  in  the  other. 


202 


THE   RESISTANCE    OF    MATERIALS. 
TABLE  III. 


• 

, 

I 

6 

4J. 

B 

"§  °° 

V    ^ 

C  g_bp 

0 

71 

ll   ' 

ft-S  | 

TEMPERATURE. 

f-a 

rt     ^ 

o    . 

•S 

tfc! 

C^-JH 

|| 

Ij 

II 

Hi 

^ 

S^ 

w 

rt 

March  6,  1837 

1.410 

*3 

78° 

June  23,  1838. 

1.457 

S  ^ 

Cold-blast. 

72° 

July  5,  1839 

1.446 

|| 

61° 

June  6,  1840  

1.445 

0.881  :  1 

50° 

November  22,  1841  . 

1.449 

'o  ^'S 

58° 

April  19,1842  

1.449 

£  -S  3 

J  p, 

Mean.  

1.442 

We  find  from  this  table,  as  from  Table  I.,  that  the  maximum 
deflection  was  observed  about  a  year  and  a  quarter  after  the 
weight  was  applied,  and  that  it  decreased  during  the  next  two 
years,  after  which  it  slightly  increased.  The  deflections  were 
the  same  at  the  two  last  observations.  These  changes  took 
place  under  the  severe  strain  of  more  than  four-fifths  of  the 
breaking  weight.  These  experiments  indicate  that  for  a  steady 
strain  which  is  less  than  three-fourths  of  the  ultimate  strength 
of  the  bar,  the  deflection  will  not  increase  progressively  until 
rupture  takes  place,  but  will  be  confined  within  small  limits. 

170.  ROEBLiNG's  OBSERVATIONS. — The  old  Monongahela 
bridge  in  Pennsylvania,  after  thirty  years  of  severe  service,  was 
removed  to  make  place  for  a  new  structure.  The  iron  which 
was  taken  from  the  old  structure  was  carefully  examined  and 
tested  by  Mr.  Eoebling,  and  found  to  be  in  such  good  condition 
that  it  was  introduced  by  him  into  the  new  bridge.* 

lie  also  found  that  the  iron  in  another  bridge  over  the  Alle- 
ghany  river  was  in  good  condition  after  forty-one  years  of 
service. 


*  Roebling's  Report  on  the  Niagara  Railroad  Bridge,   1860,  p.   1' 
Frank.  Inst.,  1860,  Vol.  LXX.,  p.  361. 


Jour. 


TORSION.  203 

OFT-REPEATED    STRAINS. 

Xearly  all  kinds  of  structures  are  subjected  to  greater  strains 
at  certain  times  than  at  others,  and  some  structures,  as  bridges 
and  certain  machines,  are  subject  to  almost  constant  changes  in 
the  strains.     Loads  are  put  on  and  removed,  and  the  operation 
constantly  repeated.     The  only  experiments  to  which  we  can 
refer  for  determining  the  effect  of  a  load  which  is  placed  upon 
a  bar  and  then  removed,  and  the  operation  of  which  was  fre- 
quently repeated,   are    those   of 
"VYm.  Faii-bairn,  made  in  I860.* 
The  beam  was  supported  at  its 
ends,  and  the  weight  which  pro- 
duced the  strain  was  raised  and 
lowered  by  means  of  a  crank  and 
pitman,  as  shown  in  Fig.  110. 

The  gearing  was  connected 
with  a  water-wheel,  which  was 
kept  in  motion  day  and  night, 
and  the  number  of  changes  of 
the  load  were  registered  by  an  automatic  counter.  The  beam 
was  20  feet  clear  span  and  16  inches  deep.  The  dimensions  of 
the  cross  section  were  as  follows : — 

Top— Plate,  4  x  %= 2.00  sq.  inches 

Angle  irons,  2  x  2  x  T5-g-=  . .  .  2.30  "         " 

Bottom— Plate,  4  x  i= f .00  "        " 

Anglelrons,2x  2x^=1.40  "         " 
Web— Plate,  15 J  x  £= 1.90  "         " 


Total  ...................  8.60 


« 


Weight  of  beam,  1  cwt.  3  qr.  3  Ibs. 

Probable  breaking  weight,  9.6  tons.      -•=-  , — ft -6 


JL    J.  wk/C*ik/ Av-/     IL/JL  \_>c*4VAJ.J.tl     TV  WAfiLAAva     U  •  \-r      fV/J.±O» 

First  Experiment. — Beam  loaded  to  J  the  breaking  weight : — 

Total  applied  load 5,809  Ibs.   ^ 

Half  the  weight  of  the  beam 434   "  _  ,    ^ 

Strain  on  the  bottom  flange 4.3  tons  p 

Margin  of   strength  by  Board   of 

rade 3.4 

.. t_ 

r.  En-   aud  Arch'.'  Jour.,  Vol.  XXIlf.,  p.  257,  and  Vol.  XXIV.,  p.  237. 


204 


THE   RESISTANCE    OF    MATERIALS. 


TABLE 

Of  the  Results  of  Experiments  made  upon  a  Beam  which  was  Supported  at  its 
Ends,  and  a  Weight  repeatedly  but  gradually  Applied  at  the  Middle. 


DATE. 

No.  of  Changes. 

Deflection  at  Cen- 
tre of  Beam. 

DATE. 

No.  of  Changes. 

O  " 

'-go 

i5 

1860. 
March  21     

0.17 

1860. 
April  13 

268,328 

0.17 

22  

10,540 

0  18 

14     

281,210 

0.17 

23  

15,610 

0.16 

17  

321,015 

0.17 

24          

27,840 

20 

343  880 

0  17 

26           .     . 

46,100 

0.16 

25     ... 

390,430 

0  16 

27     

57,790 

0.17 

27 

408,264 

0.16 

28  

72,440 

0.17 

28  

417,940 

0.16 

29  

85,960 

0.17 

May     1  .  . 

449,280 

0.16 

30 

97420 

0.17 

3   ... 

468,600 

0  16 

31            

112,810 

0.17 

5 

489,769 

0.16 

April     2  

144,350 

0.16 

7  

512,181 

0.16 

4  

165,710 

0.18 

9  

536,355 

0.16 

7  

202,890 

0.17 

11  

560,529 

0.16 

10 

235  811 

0.17 

14 

596.790 

0.16 

At  this  point,  after  half  a  million  of  changes,  the  beam  did 
not  appear  to  be  damaged.  At  first  it  took  a  permanent  set  of 
0.01  of  an  inch,  which  did  not  appear  to  increase  afterwards, 
and  the  mean  deflection  for  the  last  changes  was  less  than  for 
the  first.  For  the  last  seventeen  days  the  deflection  was  uni- 
form, but  for  the  first  seventeen  days  it  was  variable. 

The  moving  load  was  now  increased  to  one-third  the  break- 
ing weight,  =  7,406  Ibs.,  with  the  following  results  : — 


•8 
I 

I 

•s 

1 

i 

1 

• 

.S 

| 

a 

DATE. 

i 

DATE. 

J 

°-a' 

I 

|d 

•  1* 

3 

d 

1 

1860. 

1860. 

May  14  ... 

0.22 

June  7  

217,300 

15 

12623 

022 

9 

236  460 

021 

17 

36417 

0.22 

12. 

264  220 

0  21 

19  

53,770 

021 

16   

292,600 

021 

22     

85,820 

0.22 

21  

327,000 

0  22 

26 

128  300 

0  22 

23       

350  000 

0  23 

29  ... 

161  500 

022 

25  

375  650 

025 

31 

177,000 

022 

26  ;  

403  210 

023 

June    4  

194,500 

0.21 

0.23 

TORSION. 


205 


The  beam  had  now  received  1,000,000  changes  of  the  load, 
but  it  remained  uninjured.  The  moving  load  was  now  in- 
creased to  10,050  Ibs. — or  one-half  the  breaking  weight — and  it 
broke  with  5,175  changes.  The  beam  was  then  repaired  by 
riveting  a  piece  on  the  lower  flange,  so  that  the  sectional  area 
was  the  same  as  before,  and  the  experiment  was  continued. 
One  hundred  and  fifty-eight  changes  were  made  with  a  load 
equal  to  one-half  the  breaking  weight ;  and  the  load  was  then 
reduced  to  two-fifths  the  breaking  weight,  and  25,900  changes 
made.  Lastly,  the  load  was  reduced  to  one-third  the  breaking 
weight,  with  the  following  results  :— 


DATE. 

No.  of  Changes  of 
Load. 

Deflection  in  Inchs. 

DATE. 

No.  of  changes  of 
Load. 

i 
| 

I860. 
August      13 

25  900 

0  18 

1860. 
Dec    22 

929  470 

018 

16 

46,326 

29  

1,024  500 

20  

71,000 

1861. 

24  
25  

101,760 
107,000 

Jan.     9  

19             .    .. 

1,121,100 
1,278,000 

.... 

31  

135,260 

26  

1,342,800 

Sept.           1  

140,500 

Feb.     2  

1,426,000 

8. 

189  500 

11 

1  485,000 

15  

242  860 

16  

1,543,000 

22  

277,000 

23  

1,602,000 

30  

320  000 

March  2 

1  661,000 

October      6  

375,000 

9  

1,720,000 

0.18 

13  

429,000 

13  

1,779,000 

017 

20.   .. 

484000 

23     . 

1  829  000 

27  

538,000 

30  

1,885,000 

November  3  

577,800 

April   6  

1,945,000 

10 

617  800 

13 

2  000  000 

17  ... 

657,500 

20     .     . 

2  059,000 

23  

712,300 

27  

2,110,000 

December  1  

768,100 

May    4  

2,165,000 

8  

821  970 

11 

2,250,000 

15  

875,000 

018 

June  

2,727,754 

0.17 

The  piece  had  now  received  nearly  4,000,000  changes  in  all, 
but  the  2,727,000  changes  after  it  was  once  broken  and  re- 
paired did  not  injure  it.  The  changes  were  not  very  rapid. 
During  the  first  experiment  they  averaged  about  11,000  per 
day,  or  less  than  eight  per  minute,  and  during  the  last  experi- 
ment the  highest  rate  of  change  appears  to  have  been  less  than 


206  THE   KESISTANCE    OF    MATERIALS. 

eleven  per  minute,  which  is  very  slow  compared  with  the  stroke 
of  some  machines.  Tilting-hammers  often  run  from  ten  to 
twenty  times  this  speed. 


SHOCK — CRYSTALLIZATION. 

171.  SHOCKS. — When  a  weight  is  applied  to  another  body 
suddenly  it  produces  a  "  shock  "  upon  the  materials  which  com- 
pose the  bodies.  We  cannot,  practically,  tell  how  frequently  or 
with  what  force  bodies  must  come  in  contact  with  each  other  in 
order  to  produce,  a  "  shock  ;  "  but  theoretically  any  small  body 
which  is  suddenly  arrested  in  its  movement,  or  suddenly  devi- 
ated in  its  course  by  another  body,  produces  a  shock.  Mass  is 
necessary  to  the  production  of  a  shock,  and  the  masses  must  im- 
pinge upon  each  other.  If  a  force  could  manifest  itself  inde- 
pendent of  matter,  arresting  the  movement  <$.  it,  however  sud- 
denly, by  another  force,  would  not  produce  a  shock.  Also, 
changing  the  movement  of  a  mass  by  such  a  force,  however 
sudden,  will  not  produce  a  shock.  These  ideas  are  approxi- 
mately realized  in  the  movements  of  steam,  air,  and  other  gases. 
Steam  impinges  against  air  without  producing  shock,  practically 
speaking.  A  moving  piston  (in  some  machines)  is  brought  to 
rest  by  the  reaction  of  steam,  or  by  a  steam  cushion,  without 
producing  shock.  The  alternate  expansions  and  contractions  of 
a  piston-rod,  or  pitman,  or  other  similar  piece  in  steam  machin- 
ery, which  are  caused  by  the  alternate  pull  and  push  of  the 
moving  force,  do  not  produce  a  shock.  The  pieces  may  be 
"  shocked  "  on  account  of  working  with  loose  connections,  but 
that  cause  is  not  here  considered.  In  the  first  example  above 
cited  there  is  very  little  mass  in  the  moving  or  resisting  bodies ; 
in  the  next  one  the  motion  of  the  moving  mass  is  changed,  and 
may  be  brought  quite  suddenly  to  rest  by  the  action  of  a  highly 
elastic  medium  which  has  but  little  mass  ;  and  in  the  last  exam- 
ple the  particles  are  contiguous  and  are  only  slightly  moved  in 
reference  to  each  other,  as  the  forces  of  extension  and  compres- 
sion are  transmitted  through  the  bar.  The  particles  are  not 
permanently  displaced  in  reference  to  each  other,  as  they  are 
liable  to  be  by  a  blow  or  "  shock." 

Shocks   are    practically  prevented    in   many   cases   by   the 


TORSION.  207 

introduction  of  elastic  substances  which  possess  considerable 
mass.  Thus  steel,  rubber,  and  wooden  springs  in  vehicles  and 
in  certain  machines  are  familiar  examples.  But  elasticity  alone 
is  not  a  sufficient  protection,  for,  as  has  been  previously  observed, 
all  bodies  are  elastic.  When  masses  are  used  for  springs  they 
must  be  so  arranged  as  to  operate  through  a  perceptible  space 
in  bringing  the  moving  body  to  rest,  or  in  changing  its  velocity 
a  perceptible  amount. 

Springs,  however  elastic,  will  not  always  prevent  a  shock,  al- 
though they  may  greatly  relieve  it.  Thus,  the  springs  under  a 
car  will  not  prevent  the  shock  which  always  follows  when  the 
car-wheel  strikes  the  end  of  a  rail,  although  the  shock  is  not  as 
severe  as  it  would  be  if  the  body  of  the  car  were  rigidly  con- 
nected with  the  axle.  So,  too,  the  springs  between  the  buffers 
on  a  car  and  the  body  of  the  car  will  not  prevent  one  buffer 
striking  another  so  as  to  produce  a  "  shock."  In  these  cases  the 
springs  may  prevent  the  shock  from  being  transmitted  in  a 
large  degree  to  the  body  of  the  car.  The  springs  in  certain 
forge-hammers  operate  in  a  similar  way  to  prevent  a  "  shock  " 
upon  the  working  parts  of  the  machinery. 

"  Shocks  "  are  very  injurious  to  machinery,  and  hence  should, 
so  far  as  possible,  be  avoided.  All  machines  in  which  "  shocks  " 
are  necessary,  or  incidental,  or  accidental,  such  as  steam  forge- 
hammers,  morticing  machines,  stone-drilling  machines,  and  the 
like,  are  much  more  liable  to  break  than  those  that  operate  by 
a  steady  pull  and  push.  Metals  are  so  liable  to  break  under 
such  circumstances  that  many  have  supposed  that  the  internal 
structure  is  changed,  and  the  metal  becomes  more  or  less  crys- 
tallized. 

The  strength  of  the  metal  which  is  subjected  to  shocks  is  also 
greatly  modified  by  the  temperature — the  lower  the  temperature 
the  more  damaging  is  the  shock.  It  has  been  shown  in  Article 
29,  that  wrought  iron  is  somewhat  stronger  at  a  low  tempera- 
ture under  a  steady  strain  than  at  a  higher  temperature.  Not- 
withstanding this  is  contrary  to  the  "  popular  notion,"  it  has 
been  further  confirmed  by  the  very  careful  experiments  of  the 
"  Committee  appointed  by  His  Majesty  the  King  of  Sweden," 
and  reported  by  Knut  Styffe,  Director  of  the  Royal  Technolog- 


208  THE    RESISTANCE    OF   MATERIALS. 

ical  Institute  of  Stockholm.*  Their  first  conclusion  was  : — f 
"  The  absolute  strength  (tenacity)  of  iron  and  steel  is  not  dimin- 
ished by  cold,  but  that  even  at  the  lowest  temperature  which 
ever  occurs  in  Sweden,  it  is  at  least  as  great  as  at  the  ordinary 
temperature — or  60°  Fahr." 

These  results  are  confirmed  by  the  more  recent  experiments 
of  Joule  of  England,  and  by  several  other  experimenters. 

But  it  is  generally  supposed  that  machinery,  railroad  iron, 
tyres  on  locomotives,  and  tools,  break  much  more  frequently 
with  the  same  usage  when  very  cold  than  they  do  when  warm. 
Is  this  a  mere,  notion  ? 

Is  it  because  breakages  are  more  annoying  in  cold  than  in 
warm  weather,  and  hence  make  a  more  lasting  impression  upon 
the  minds  of  those  who  have  to  deal  with  them ;  so  that  they 
think  they  occur  more  frequently  ?  Impressions  are  not  safe 
guides  in  scientific  investigations.  Our  observations  on  the  use  of 
out-door  machinery  in  cold  and  warm  weather  lead  us  to  believe 
that  they  do  break  much  more  frequently  with  the  same  usage 
in  winter  than  in  summer.  The  same  fact,  in  regard  to  the 
breaking  of  rails  on  railroads,  was  admitted  by  Styffe ;  but 
after  arriving  at  the  conclusion  which  he  did  in  regard  "to  the 
effect  of  cold  upon  the  absolute  strength  of  iron,  he  concluded 
that  the  cause  of  the  more  frequent  breakages  was  due  to  the 
more  rigid  and  non-elastic  foundation  caused  by  the  frozen 
ground.  But  Sandberg,  the  translator  of  Styffe's  work,  thought 
that  iron  when  subjected  to  shocks  might  not  give  the  same 
relative  strength  at  different  temperatures  that  it  would  when 
subjected  to  a  steady  strain.  He  therefore  instituted  another 
series  of  experiments  to  satisfy  himself  upon  this  important 
point,  and  aid  in  solving  the  problem.  The  following  is  an 
abstract  of  his  report  :• — 

The  supports  for  the  rails  in  the  experiments  wrere  two  large 
granite  blocks  which  rested  upon  granite  rocks  in  their  native 
bed.  The  rails  were  supported  near  their  ends  on  these  blocks. 
They  were  broken  by  a  ball  which  weighed  9  cwt,  which  was 
permitted  to  fall  five  feet  the  first  blow,  and  the  height  increased 


*  The  Elasticity,  Extensibility,  and  Tensile  Strength  of  Iron  and  Steel.     By 
Knut  Styffe.     Translated  by  Christer  P.  Sandberg,  London, 
f  Ibid.,  p.  111. 


EFFECT   OF    SHOCKS.  209 

one  foot  at  each  succeeding  fall,  and  the  deflection  measured 
after  each  impact.  A  small  piece  of  wrought  iron  was  placed 
on  the  top  of  each  rail  to  receive  the  blow,  so  as  to  concentrate 
its  effect. 

The  rail  was  thus  broken  into  two  halves,  and  each  part  was 
afterwards  broken  at  different  temperatures.  As  the  experi- 
ments were  not  made  till  the  latter  part  of  the  winter,  the 
lowest  temperature  secured  was  only  10°  Fahr.  Fourteen  rails 
were  tested: — Seven  of  which  were  from  Wales;  five  from 
France  ;  and' two  from  Belgium.  From  these  the  experimenter 
drew  the  following  conclusions  : — * 

1.  "  That  for  such  iron  as  is  usually  employed  for  rails  in  the 
three  principal  rail-making  countries  (Wales,  France,  and  Bel- 
gium), the  breaking  strain,  as  tested  by  sudden  Hows  or  shocks, 
is  considerably  influenced  by  cold ;  such  iron  exhibiting  at  10° 
F.,  only  one-third  to  one-fourth  of  the  strength  which  it  pos- 
sesses at  84°  F. 

2.  u  That  the  ductility  and  flexibility  of  such  iron  is  also 
much  affected  by  cold,  rails  broken  at  10°  F.  showing  on  an  av- 
erage, a  permanent  deflection  of  less  than  one  inch,  whilst  the 
other  halves  of  the  same  rails,  broken  at  84°  F.,  showed  less  than 
four  inches  before  fracture." 

This  seems  to  be  the  fairest  and  most  conclusive  experiment 
upon  this  point  that  we  have  met  with,  but  it  is  not  satisfactory 
to  all,  or  else  they  are  ignorant  of  the  experiment,  for  there  has 
been  of  late  considerable  discussion  upon  the  subject  in  the 
scientific  journals.  Some  take  the  experiments  of  Fairbairn 
and  Joule  as  conclusive  upon  the  point,  and  attribute  the  cause 
of  the  failures,  in  many  cases,  to  an  inferiority  of  the  iron,  and 
in  the  case  of  tyres  to  an  over-stretching  of  the  metal  when  it 
is  put  on  the  wheel.  By  many,  the  presence  of  phosporous 
is  considered  especially  detrimental  to  iron  which  is  subject- 
ed to  shocks  in  cold  weather.  But  until  the  fact  is  estab- 
lished that  cold  iron  is  weaker  than  warm  iron,  when  subjected 
to  shocks,  it  is  worse  than  useless  to  speculate  upon  the  cause. 
More  experiments  are  needed  on  this  point,  in  which  the  quality 
of  the  metal,  and  all  the  conditions  of  the  experiments  should  be 
definitely  known. 

*  Styffe's  work  on  iron  and  steel,  translated  by  Sandberg,  p.  157. 
14 


i 


210  THE   RESISTANCE    OF   MATERIALS. 

The  following  experiments,  by  John  A.  Roebling,*  bear  upon 
this  subject,  although  they  are  not  conclusive,  for  it  is  not  re- 
ported that  the  same  metal  was  tested  when  warm  :  — 

"  The  samples  tested  were  about  one  foot  long,  and  were  re- 
duced at  the  centre  to  exactly  three-fourths  of  an  inch  square, 
and  their  ends  left  larger,  were  welded  to  heavy  eyes,  making 
in  all  a  bar  three  feet  long.  These  were  covered  with  snow  and 
ice,  and  left  exposed  several  days  and  nights.  Early  in  the 
morning,  before  the  air  grew  warmer,  a  sample  inclosed  in  ice, 
was  put  into  the  testing-machine  and  at  once  subjected  to  a 
strain  of  26,000  pounds,  the  bar  being  in  a  vertical  position,  and 
left  free  all  around.  The  iron  was  capable  of  resisting  70,000 
Ibs.  to  80,000  Ibs.  per  square  inch.  A  stout  mill-hand  struck  the 
reduced  section  of  the  piece,  horizontally,  as  hard  as  he  could, 
with  a  billet  one  and  a  half  inches  in  diameter  and  two  feet 
long.  The  samples  resisted  from  three  to  one  hundred  and 
twenty  blows.  With  a  tension  of  20,000  Ibs.  some  good  sam- 
ples resisted  300  blows  before  breaking." 

The  finest  and  best  qualities  of  iron,  or  those  that  have  the 
highest  coefficient  elasticity  will  resist  vibration  best.  It  is 
generally  supposed  that  good  iron  will  resist  concussions  much 
better  than  steel.  Sir  William  Armstrong,  of  England,  says  :  — 
"  The  conclusion  at  which  I  have  long  since  arrived,  and  which 
I  still  maintain,  is,  that  although  steel  has  much  greater  tensile 
strength  than  wrought  iron,  it  is  not  as  well  adapted  to  resist 
concussive  strains.  It  is  impossible,  then,  that  the  vibratory 
action  attending  concussion  is  more  dangerous  to  iron  than  to 
steel.  The  want  of  uniformity  is  another  serious  objection  to 
the  general  use  of  steel  in  such  cases."  This  has  been  used 
as  an  argument  against  the  use  of  steel  rails,  but  practically 
this  has  proved  not  to  be  a  serious  difficulty.  So  many  ele- 
ments must  be  considered  in  the  use  of  steel  rails,  aside  from 
fracture,  that  the  problem  must  be  solved  by  itself,  and  princi- 
pally upon  other  grounds. 


CRYSTALLIZATION.  —  A  crystal  is  a  homogeneous  in- 
organic solid,  bounded  by  plane  surfaces,  systematically  arranged. 
The  quartz  crystal  is  a  familiar  example.  Different  substances 


Jour.  Frank.  Inst.,  vol.  xl.,  3d  series,  p.  361. 


CRYSTALLIZATION. 

crystallize  in  forms  which  are  peculiar  to  themselves.  Metals, 
under  certain  circumstances,  crystallize ;  and  if  they  are  broken 
when  in  this  condition  the  fracture  shows  small  plane  surfaces, 
which  are  the  faces  of  the  crystals.  It  is  found  in  all  cases 
that  crystallized  iron  is  weaker  than  the  same  metal  in  its  or- 
dinary state.  By  its  ordinary  state  we  mean  that  wrought 
iron  is  fibrous,  and  cast  iron  and  steel  are  granular  in  their 
appearance. 

Iron  crystallizes  in  the  cubical  system.*  Wholer,  in  break- 
ing cast-iron  plates  readily  obtained  cubes,  when  the  iron  had 
long  been  exposed  to  a  white  heat  in  the  brickwork  of  an  iron 
smelting  furnace. 

Augustine  found  cubes  in  the  fractured  surface  of  gun  barrels 
which  had  long  been  in  use. 

Percy  found  on  the  surface  and  interior  of  a  bar  of  iron, 
which  had  been  exposed  for  a  considerable  time  in  a  pot  of 
glass-making  furnace,  large  skeleton  octahedra.  (He  seems 
to  differ  from  the  preceding  in  regard  to  the  •  form  of  the 
crystals.) 

Prof.  Miller,  of  Cambridge,  found  Bessemer  iron  to  consist 
of  an  aggregation  of  cubes,  ij  -,, .  ,  ..?  » 

Mallet  says : — "  The  plans,  of  crystallization  group  themselves 
perpendicular  to  the  external  surfaces." 

Bar  iron  will  become  crystalline  if  it  is  exposed  for  a  long 
time  to  a  heat  considerably  below  fusion.  Hence  we  see  why 
large  masses  which  are  to  be  forged  may  become  crystalline,  on 
account  of  the  long  time  it  takes  to  heat  the  mass.  Forging 
does  not  destroy  the  crystals,  and  forging  iron  at  too  low  a 
temperature  makes  it  tender,  while  steel  at  too  high  a  tempera- 
ture is  brittle.  The  presence  of  phosphorous  facilitates  crystal- 
lization. Time,  in  the  process  of  breaking  iron,  will  often 
determine  the  character  of  the  fracture.  If  the  fracture  is 
slow,  the  iron  will  generally  appear  fibrous  ;  but  if  it  be  quick, 
it  will  appear  more  or  less  crystalline.  Many  mechanics  have 
noticed  this  result.  At  Shoeburyness  armor-plates  were  shat- 
tered like  glass  under  the  impact  of  shot  at  a  velocity  of  1,200 
feet  to  1,600  feet  per  second.  The  iron  was  good  fibrous  iron. 

Many  engineers  are  of  the  opinion  that  oft-repeated  and  long- 

*  See  Osborn's  Metallurgy,  pp.  83-86. 


212  THE   RESISTANCE    OF    MATERIALS. 

continued  shocks  will  change  fibrous  to  crystalline  iron.  Opin- 
ions, however,  are  divided  upon  this  subject.  In  view  of  the 
immense  amount  of  machinery  and  other  constructions,  parts  of 
which  are  constantly  subjected  to  shocks,  the  importance  of  the 
subject  can  hardly  be  overestimated. 

William  Fairbairn  says: — *  "We  know  that  in  some  cases 
wrought  iron  subjected  to  continuous  vibration  assumes  a  crys- 
talline structure,  and  that  then  the  cohesive  powers  are  much 
deteriorated ;  but  we  are  ignorant  of  the  causes  of  this  change." 

The  late  Robert  Stephenson  f  referred  to  a  beam  of  a  Cor- 
nish engine  which  received  a  shock  eight  or  ten  times  a  minute, 
equal  to  about  fifty  tons,  for  a  period  of  twenty  years  without 
apparent  change.  These  shocks  were  not  very  frequent,  and 
would  not  be  considered  as  detrimental  as  if  they  occurred  a 
few  times  each  second.  He  also  says : — "  The  connecting-rod  of  a 
certain  locomotive  engine  that  had  run  50,000  miles,  and  re- 
ceived a  violent  jar  eight  times  per  second,  or  25,000,000  vibra- 
tions, exhibited  no  alteration."  In  all  the  cases  investigated  by 
him  of  supposed  change  of  texture,  he  knew  of  no  single  in- 
stance where  the  reasoning  was  not  defective  in  some  important 
link.  These  are  not  fair  examples  of  shocks,  as  the  vibrations 
referred  to  seem  to  be  only  changes  from  tension  to  compression, 
and  the  reverse. 

Mr.  Brunei  accepted  the  theory  of  molecular  change,  for  a 
time,  as  due  to  shocks,  but  afterwards  expressed  great  doubts 
as  to  its  correctness,  and  thought  that  the  appearance  depended 
more  upon  the  manner  of  breaking  the  metal  than  upon  any 
molecular  change. 

Fairbairn  has  speculated  a  little  upon  the  probable  cause  of 
the  internal  change  when  it  takes  place.  In  his  evidence  before 
the  Commissioners  appointed  to  inquire  into  the  application 
of  iron  to  railway  structures,  he  says  : — "  As  regards  iron  it  is 
evident  that  the  application  and  abstraction  of  heat  operates 
more  powerfully  in  effecting  these  changes  than  probably  any 
other  agency  ;  and  I  am  inclined  to  think  that  we  attribute  too 
much  influence  to  percussion  and  vibration,  and  neglect  more 
obvious  causes  which  are  frequently  in  operation  to  produce 

*  Civ.  Eng.  and  Arch.  Jour.,  vol.  iii.  p.  257. 
f  Am.  R.  Times,  March  6,  1869,  Boston. 


CRYSTALLIZATION.  213 

the  change.  For  example,  if  we  take  a  bar  of  iron  and  heat  it 
red  hot,  and  then  plunge  it  into  water,  it  is  at  once  converted 
into  a  crystallized  instead  of  a  fibrous  body  ;  and  by  repeating 
this  process  a  few  times,  any  description  of  malleable  iron  may 
be  changed  from  a  fibrous  to  a  crystalline  structure.  Vibration, 
when  produced  by  the  blows  of  a  hammer  or  similar  causes, 
such  as  the  percussive  action  upon  railway  axles,  I  am  willing 
to  admit  is  considerable;  but  I  am  not  prepared  to  accede  to 
the  almost  universal  opinion  that  granulation  is  produced  by 
those  causes  only.  I  am  inclined  to  think  that  the  injury  done 
to  the  body  is  produced  by  the  weight  of  the  blow,  and  not  by 
the  vibration  caused  by  it.  If  we  beat  a  bar  with  a  small  ham- 
mer, little  or  no  effect  is  produced ;  but  the  blows  of  a  heavy 
one,  which  will  shake  the  piece  to  the  centre,  will  probably  give 
the  key  to  the  cause  which  renders  it  brittle,  but  probably  not 
that  which  causes  crystallisation.  The  fact  is,  in  my  opinion, 
we  cannot  change  a  body  composed  of  a  fibrous  texture  to  that 
of  a  crystalline  character  by  a  mechanical  process,  except  only 
in  those  cases  where  percussion  is  carried  to  the  extent  of  pro- 
ducing considerable  increase  of  temperature.  We  may,  how- 
ever, shorten  the  fibres  by  continual  bending,  and  thus  render 
the  parts  brittle,  but  certainly  not  change  the  parts  which  were 
originally  fibrous  into  crystals. 

For  example,  take  the  axle  of  a  car  or  locomotive  engine, 
which,  when  heavily  loaded  and  moving  with  a  high  velocity, 
is  severely  shocked  at  every  slight  inequality  of  the  rails.  If, 
under  these  circumstances,  the  axle  bends — however  slightly — it 
is  evident  that  if  this  bending  be  continued  through  many 
thousand  changes,  time  only  will  determine  when  it  will  break. 
Could  we,  however,  suppose  the  axle  so  infinitely  rigid  as  to  re- 
sist the  effects  of  percussion,  it  would  then  follow  that  the  in- 
ternal structure  of  the  iron  will  not  be  injured,  nor  could  the 
assumed  process  of  crystallization  take  place." 

The  late  John  A.  Roebling,  the  designer  and  constructor  of 
the  Niagara  Railway  Suspension  Bridge,  in  his  report  on  that 
structure  in  I860,*  says  he  has  given  attention  to  this  subject 
for  years,  and  as  the  result  of  his  observation,  study,  and  experi- 
ment, gives  as  his  view  that  "  a  molecular  change,  or  so-called 

*  Jmr.  Frank.  Inst,  vol.  xL,  3d  series,  p.  361. 


214  THE   RESISTANCE    OF   MATERIALS. 

granulation  or  crystallization,  in  consequence  of  vibration  or 
tension,  or  both  combined,  has  in  no  instance  been  satisfactorily 
proved  or  demonstrated  by  experiment."  "  I  further  insist  that 
crystallization  in  iron  or  any  other  metal  can  never  take  place  in 
a  cold  state.  To  form  crystals  at  all,  the  metal  must  be  in  a 
highly  heated  or  nearly  molten  state."  Notwithstanding  these 
positive  statements,  he  still  hesitates  to  express  a  decided  opinion 
which  will  cover  the  whole  field  of  investigation.  Still  further 
on  he  states  that  he  is  witnessing  the  fact  daily  that  vibration 
and  tension  combined  will  greatly  affect  the  strength  of  iron 
without  changing  its  fibrous  texture.  Wire  ropes  and  iron  bars 
will  become  weakened  as  the  vibration  and  tension  to  which  they 
are  subjected  increase. 

Certain  machines  in  which  the  working  parts  are  subjected 
to  frequent  shocks,  more  or  less  severe,  are  constantly  failing, 
and  the  general  impression  is  that  the  failure  is  due  to  crystal- 
lization. In  speaking  of  the  rock-drilling  engines  used  in 
Hoosac  Tunnel,  Mass.,  which  were  driven  by  compressed  air, 
the  committee  say :  * — "  Gradually  they  begin  to  fail  in  strength ; 
the  incessant  and  rapid  blows — counted  by  millions — to  which 
they  are  subjected,  appearing  to  granulate  or  disintegrate  por- 
tions of  the  metals  composing  them."  Having  had  some  experi- 
ence with  this  class  of  machines,  I  know  something  of  the  diffi- 
culties which  surround  them.f  During  the  winter  of  1866-7 
my  assistant  in  the  University,  Professor  S.  W.  Robinson,  and 
myself  made  several  experimental  machines,  in  the  use  of  which 
we  learned  many  essential  conditions  which  must  be  observed 
in  order  to  avoid  frequent  breakages  in  the  use  of  iron  which 
is  subjected  to  frequent  and  long-continued  shocks.  As  first 
designed,  the  breakages  in  the  several  working  parts  were  ex- 
ceedingly numerous,  the  remedy  for  which  wras  not  in  making 
those  parts  larger  and  stronger,  for  that  only  aggravated  the 
evil  in  most  cases,  but  in  arranging  the  moving  parts  so  that 
they  would  be  moved  and  brought  to  rest  with  as  little  shock  as 
possible,  and  then  making  them  as  light  as  possible  consistent 


*  Annual  Report  of  the  Commissioners  on  the  Troy  and  Greenfield  Railroad 
and  Hoosac  Tunnel.     House  Doc. ,  No.  30,  p.  5,  Boston,  Mass. 

f  Notice  of,  by  B.  H.  Latrobe,  C.  E.    Sen.  Doc.  No.  20,  1868,  p.  31,  Boston, 


EFFECT   OF    SHOCKS.  215 

with  strength.  But  at  the  rapid  rate  at  which  we  ran  them, 
which  was  from  300  to  over  500  blows  per  minute,  it  was  not 
easy  to  comply  with  these  conditions.  No  especial  difficulty 
was  experienced  from  the  general  shock  caused  by  the  striking 
of  the  tool  upon  the  rock ;  but  the  chief  difficulty  arose  from 
the  blow  or  shock  to  which  the  working  pieces  were  directly 
subjected  in  operating  them.  At  last,  however,  according  to 
the  report  of  the  superintendent  of  the  Marmora  Iron  Mine, 
Out.,  at  which  place  they  have  been  in  use  for  some  time,  we  so 
far  overcame  all  the  difficulties  as  to  make  it  a  decidedly  prac- 
tical machine.  In  the  experiments  we  found  that  it  was  a  bad 
condition  to  subject  a  piece  to  a  blow  crosswise  of  its  length ; 
that  is,  perpendicular  to  its  length.*  It  was  also  found  that  a 
piece  struck  obliquely  would  sustain  a  much  greater  number  of 
blows  than  if  struck  perpendicularly.  Many  pieces  evidently 
broke  slowly,  and  was  analogous  to  breaking  a  piece  of  tough 
iron  on  an  anvil  by  comparatively  light  blows.  If  the  blows 
were  so  severe  as  to  start  a  crack  in  the  piece,  it  would  ulti- 
mately break  if  the  blows  were  continued  sufficiently  long. 
Several  of  the  broken  pieces  were  critically  examined  to  see  if 
they  were  crystallized,  but  there  were  no  indications  of  any 
change  in  the  internal  structure  of  the  metal. 

All  sharp  angles  in  pieces  which  are  shocked  should,  if  pos- 
sible, lie  avoided  /  for  in  the  process  of  manufacture  they  are 
liable  to  be  rendered  weaker  at  such  points,  and  if  they  are 
equally  strong  so  far  as  manufacture  is  concerned,  a  greater 
strain,  at  the  instant  they  are  shocked,  is  liable  to  fall  at  such 
points,  thus  rendering  them  relatively  weaker  there.  At  least 
it  is  found  that  such  pieces  are  more  liable  to  break  at  the 
angle.  Hence,  in  the  construction  of  direct-action  rock-drills, 
direct-action  steam-hammers,  and  similar  percussion  machines, 
the  steam  piston  is  not  only  generally  made  solid  with  the  rod, 
but  it  is  connected  with  it  by  a  curve.  In  other  words,  the  rod 
is  more  or  less  gradually  enlarged  into  a  piston.  At  first,  much 
difficulty  was  experienced  in  this  regard  with  steam-hammers, 

*  I  have  seen  it  published  that  a  small  hammer  was  made  to  strike  a  blow 
upon  the  side  of  a  bar  which  was  suspended  vertically,  the  blows  being  repeated 
night  and  day  for  nearly  a  year,  when  the  bar  broke  ;  but  as  the  force  of  the 
blow  and  size  of  the  bar  were  not  given,  I  have  thought  that  the  statement 
was  too  indefinite  to  be  of  any  scientific  value. 


216 


THE   RESISTANCE    OF    MATERIALS. 


for  in  some  cases  the  piston  broke  away  from  the  rod,  and 
slipped  down  over  it. 

In  certain  steam  forge-hammers  the  piston-rod  is  liable  to 
break  where  it  joins  the  hammer.  In  this  case  the  worst  pos- 
sible arrangement  is  to  make  a  rigid  connection  between  the 
hammer  and  rod.  The  author  once  saw  a  rod  three  and  one- 
half  inches  in  diameter  nicely  fitted  into  a  conical  hole  in  a  400- 
pound  hammer,  the  taper  of  the  rod  and  hole  being  slight,  so 
that  it  would  hold  by  friction  when  once  driven  into  place. 
The  connection  was  practically  rigid.  The  rod  broke  twice  with 
ordinary  use  inside  of  twenty  days.  Probably  it  would  have 
lasted  a  long  time  if  the  blow  could  always  have  been  exactly 
central ;  but  in  ordinary  use  it  would  very  naturally  be  sub- 
jected to  cross  strains  by  making  a  blow  when  the  material  was 
under  one  edge  of  the  hammer.  By  a  repetition  of  these  cross 
strains,  rupture  might  have  been  produced  without  any  crystal- 
lization. 


Fia.  112. 

This  difficulty  is  practically  overcome  in  several  ways,  one  of 
the  most  common  of  which  is  to  place  blocks  of  wood  or  other 
slightly  elastic  body  between  the  end  of  the  piston-rod  and  the 
hammer.  Morrison  overcame  the  difficulty  by  making  the  rod 
so  large  that  it  (the  rod)  became  the  hammer,  and  a  small  block 
of  iron  or  steel  was  fitted  into  the  forward  end  of  the  rod  to 
serve  as  a  face  for  the  hammer. 

A  Mr.  Webb,  of  England,  proposed  to  overcome  the  difficulty 
hy  a  device  which  is  shown  in  Fig.  112. 


CRYSTALLIZATION.  217 

Referring  to  the  figure,  it  will  be  seen  that  the  piston-rod, 
which  is  for  the  main  part  of  its  length  4  in.  in  diameter,  is  en- 
larged at  the  lower  end  to  6J  in.  in  diameter,  and  is  shaped 
spherically.  This  spherical  portion  of  the  rod  is  embraced  by 
the  annealed  steel  castings,  B  B,  which  are  secured  in  their 
place  in  the  hammer-head  by  the  cotters,  A,  and  the  whole 
thus  forms  a  kind  of  ball-and-socket  joint,  which  permits  the 
hammer  head  to  swivel  slightly  on  the  rod  without  straining 
the  latter.  Mr.  Webb  first  applied  this  form  of  hammer-rod 
fastening  to  a  five-ton  Nasmyth  hammer  with  a  4  in.  rod.  With 
the  old  mode  of  attachment,  with  a  cheese  end,  this  hammer 
broke  a  rod  every  three  or  four  weeks  when  working  steel,  while 
a  rod  with  the  ball-and-socket  joint,  which  was  put  in  in  No- 
vember, 1867,  has  been  working  ever  since,  that  is,  to  some  time 
in  1869,  without  giving  any  trouble.  The  inventor  has  also  ap- 
plied a  rod  thus  fitted  to  a  five-ton  Thwaites  and  Carbutt's  ham- 
mer with  equal  success. 

These  examples  seem  to  indicate  that  if  iron  is  crystallized  on 
account  of  shocks,  the  progress  of  the  change  may  be  slackened 
by  a  judicious  arrangement  of  the  pieces  and  by  proper  connec- 
tions. But  it  does  not  follow  that  because  a  metal  breaks  so 
frequently  when  subjected  to  shock,  that  it  has  become  crystal- 
lized. It  has  been  observed  by  those  who  have  made  the  experi- 
ment, that  a  piece  of  bar  iron  which  is  broken  by  heavy  blows, 
when  the  piece  is  so  supported  as  to  bound  with  each  stroke, 
will  present  a  crystalline  fracture  ;  but  if  the  same  bar  be  broken 
by  easy  blows  near  the  place  of  the  former  fracture,  it  will  pre- 
sent a  fibrous  texture  ;  showing  that  in  the  former  case  the  in- 
ternal structure  was  not  changed,  unless  it  were  in  the  immedi- 
ate vicinity  of  the  fracture.  In  such  cases  the  appearance  of 
the  surface  of  the  fracture  does  not  indicate  the  true  state  of 
the  internal  structure.  One  reason  why  metals  fail  which  are 
subjected  to  concussions,  without  crystallizing,  is  ; — an  excessive 
strain  is  brought  upon  some  point,  thus  impairing  the  elasticity 
and  weakening  the  resisting  powers,  in  which  case,  if  the  strains 
be  repeated  sufficiently  long,  rupture  must  ultimately  take  place. 
Or,  if  the  concussion  be  sufficiently  severe  and  local,  it  may  dis- 
place the  particles,  and  thus  begin  a  fracture.  This  frequently 
takes  place  in  the  case  of  anvils,  hammers,  hammer-blocks,  and 
the  like. 


218  THE    RESISTANCE    OF   MATERIALS. 

If  a  bar  is  bent  by  a  blow  sufficiently  great  to  produce  a  set,  and 
the  bar  be  bent  back  by  another  blow,  and  so  on,  the  bar  being 
bent  alternately  to  and  fro,  rupture  would  probably  take  place 
at  some  time,  however  remote.  It  is  often  difficult  to  determine 
the  strain  which  falls  at  a  particular  point  of  a  piece  when  it  is 
subjected  to  a  shock,  but  if  we  could  determine  its  exact  amount, 
we  might  find  it  to  be  sufficiently  large  to  account  for  rupture 
by  shocks,  without  considering  any  mysterious  change  in  the 
internal  structure  of  the  metal. 

Note.  Since  the  above  was  written,  several  articles  have 
appeared  in  the  scientific  journals,  giving  the  results  of  obser- 
vations and  experiments  upon  the  strength  of  iron  at  low  tem- 
peratures, and  they  all  confirm  the  position  above  stated, — that 
iron  will  not  resist  shocks  as  effectually  at  very  low  temperatures 
as  it  will  at  ordinary  temperatures. 


LIMITS  OF  SAFE  LOADING  OF  MECHANICAL  STRUCTURES.        219 


CHAPTER  X. 

LIMITS  OF  SAFE  LOADING  OF  MECHANICAL  STRUCTURES. 

173.—  RISK  AND  sAFETY.-We  have  now  considered  the 
breaking-strength  of  materials  under  a  variety  of  conditions, 
and  also  the  changes  produced  upon  them  when  the  strains  are 
within  the  elastic  limits.  In  a  mechanical  structure,  in  which 
a  single  piece,  or  a  combination  of  pieces,  are  required  to  sus- 
tain a  load,  it  is  desirable  to  know  how  small  the  piece,  or  the 
several  pieces,  may  be  made  to  sustain  a  given  load  safely  for 
an  indefinite  time  ;  or,  how  much  a  given  combination  will  sus- 
tain safely.  The  nature  of  the  problem  is  such  that  an  exact 
limit  cannot  be  fixed.  Materials  which  closely  resemble  each 
other  do  not  possess  exactly  the  same  strength  or  stiffness  ;  and 
the  conditions  of  the  loading  as  to  the  amount  or  manner  in 
which  it  is  to  be  applied,  may  not  be  exactly  complied  with. 
Exactness,  then,  is  not  to  be  sought  ;  but  it  is  necessary  to  find  a 
limit  below  which,  in  reference  to  the  structure,  or  above  which, 
in  reference  to  the  load,  it  is  not  safe  to  pass. 

It  is  evident  that  to  secure  an  economical  use  of  the  material 
on  the  one.  hand,  and  ample  security  against  failure  on  the  other, 
the  limit  should  be  as  definitely  determined  as  the  nature  of  the 
problem  will  admit  ;  but  in  any  case  we  should  incline  to  the 
side  of  safety.  No  doubt  should  be  left  as  to  the  stability 
of  the  structure.  There  is  no  economy  in  risk  in  permanent 
structures.  Risk  should  be  taken  only  in  temporary,  or  experi- 
mental, structures  ;  or  where  risk  cannot,  from  the  nature  of 
the  case,  be  avoided. 


.  ABSOLUTE  ITIOWI  I,AS  OF  SAFETY.  —  In  former  times, 
one  of  the  principal  elements  which  was  used  for  securing 
safety  in  a  structure,  was  to  assume  some  arbitrary  value  for 
the  resistance  of  the  material,  such  value  being  so  small  that 


220 


THE   RESISTANCE   OF   MATERIALS. 


the  material  could,  in  the  opinion  of  the  engineer,  safely  sus- 
tain it.  This  is  a  convenient  mode,  but  very  unphilosophical, 
although  still  extensively  used.  The  plan  was  to  determine,  as 
nearly  as  possible,  what  good  materials  would  sustain  for  a  long 
period,  and  use  that  value  for  all  similar  materials.  But  it  is  evi- 
dent, from  what  has  been  said  in  the  preceding  pages,  that  some 
materials  will  sustain  a  much  larger  load  than  the  average, 
while  others  will  not  sustain  nearly  so  much  as  the  average. 
In  all  such  cases  the  proper  value  of  the  modulus  can  only  be 
determined  by  direct  experiment.  In  all  important  structures 
the  strength  of  the  material,  especially  iron  and  steel,  should  be 
determined  by  direct  experiment. 

The  following  values  are  generally  assumed  for  the  modulas 
of  safety. 

Pounds  per  square  inch. 

Wrought,  iron  for  tension  or  compression,  from 10,000  to  12,000 

Cast  iron,  for  tension,  from 3,000  to    4,000 

Cast  iron,  for  compression,  from 15,000  to  20,000 

Wood,  tension  or  compression,  from 850  to    1,200 

f granite,  from 400  to    1,200 

!  quartz,  from 1,200  to     2,000 

Stone,  expression    j  g4andstone?  from 300  to       600 

[limestone,  from 800  to    1,200 

The  practice  of  French  engineers,*  in  the  construction  of 
bridges,  is  to  allow  3.8  tons  (gross)  per  inch  upon  the  gross  sec- 
tion, both  for  tension  and  compression  of  wrought  iron. 

The  Commissioners  on  Hailroad  Structures,  England,  estab- 
lished the  rule  that  the  maximum  tensile  strain  upon  any  part 
of  a  wrought  iron  bridge  should  not  exceed  five  tons  (gross) 
per  square  inch,  f 

In  most  cases  the  effective  section  is  the  section  which  is  sub- 
jected to  the  strain  considered. 

1 75.  FACTOR  OF  SAFETY. — The  next  mode,  and  one  which 
is  also  largely  in  use,  is  to  take  a  fractional  part  of  the  ultimate 
strength  of  the  material,  for  the  limit  of  safety.  The  recipro- 
cal of  this  fraction  is  called  the  factor  of  safety.  It  is  the  ratio  of 
the  ultimate  strength  to  the  computed  strain,  and  hence  is  the 


*  Am.  R.  R.  Times,  1871,  p.  6. 

f  Civ.  Eng.  and  Arch.  Jour.     Vol.  xxiv.,  p.  327. 


LIMITS  OF  SAFE  LOADING  OF  MECHANICAL  STRUCTURES.        221 

factor  by  which  the  computed   strain  must  be  multiplied  to 
equal  the  actual  strength  of  the  material,  or  of  the  structure. 

Experiments  and  theory  combine  to  teach  that  the  factor  of 
safety  should  not  be  taken  as  small  as  2.  See  articles  19,  166, 
167,  and  168. 

Beyond  this  the  factor  is  somewhat  arbitrarily  assumed,  de- 
pending upon  the  ideas  of  the  engineer.  For  instance,  the  fol- 
lowing values  were  given  to  the  -Commissioners  on  Railway 
Structures,  in  England.* 

Factors. 

Messrs.  May  and  Grissel '. 3 

Mr.  Brunell 3  to  5 

Messrs.  Rasbrick,  Barlow  and  others 6 

Mr.  Hawkshaw 7 

Mr.  Gly  n 10 

The  following  values  are  also  given  by  others  : — 

Factors. 

Bow,  for  wrought-iron  beams 3.5 

Weisbach,  for  wrought  iron  f 3  to  4 

Yicat,  for  wire  suspension  bridges more  than  4 

Eankine,  for  wire  bridges  j  stead?  s,train 3  to  4 

I  moving  load 6  to  8 

Fink,  iron  -  truss   bridges  \  *or  PO8ts  and  ^"f8 5  to  f 

(  for  cast-iron  chords 7 

Fairbairn,  for  cast  iron  beams  J 5to6 

C.  Shaler  Smith,  compression  of  cast  iron 5 

Rankine  and  others,  for  cast-iron  beams 4  to  6 

Mr.  Clark  in  Quincy  Bridge,  lower  chord 6  to  7 

Washington  A.  Roebling,  for  suspension  cables 6 

Morin,  Vicat,  Weisbach,  Rondelet,  Navier,  Barlow,  and 
many  others  say  that  for  a  wooden  frame  it  should  not 

be  less  than 10 

For  stone,  for  compression : 10  to  15 

From  the  experiments  which  are  recorded  in  Article  170, 
Fairbairn  deduced  the  following  conclusions  in  regard  to  beams 

*  Civ.  Eng.  and  Arch.  Jour. ,  Vol.  xxiv,  p.  327. 
f  Weisbach,  Mech.  and  Eng.  Vol.  1,  p.  201. 
J  Fairbairn,  Cast  and  Wrought  Iron,  p.  58. 


222  THE   RESISTANCE   OF   MATERIALS. 

and  girders,  whether  plain  or  tubular.  *  "  The  weight  of  the 
girder  and  its  platform  should  not  in  any  case  exceed  one-fourth 
the  breaking  weight,  and  that  only  one-sixth  of  the  remaining 
three-fourths  of  the  strength  should  be  used  by  the  moving  load." 
According  to  this  statement  the  maximum  load,  including  the 
live  and  dead  load,  may  equal,  but  should  not  exceed, 

i  +  i  of  i  =  f 

of  the  breaking  load.  Hence  the  factor  of  safety  must  not  be  less 
than  2.66  when  the  above  conditions  are  fulfilled.  This  value 
is,  however,  evidently  smaller  than  is  thought  advisable  by  most 
engineers. 

The  rule  adopted  by  the  Board  of  Trade,  England,  for  rail- 
road bridges  is  f  "  to  estimate  the  strain  produced  by  the  greatest 
weight  which,  can  possibly  come  upon  a  bridge  throughout  every 
part  of  the  structure  which  should  not  exceed  one-fifth  the  ulti- 
mate strength  of  the  metal"  They  also  observed  that  ordinary 
road  bridges  should  be  proportionately  stronger  than  ordinary 
railroad  bridges. 


RATIONAL  LIMIT  OF  SAFETY.  —  It  is  evident  that 
materials  may  be  strained  any  amount  within  the  elastic  limit. 
Their  recuperative  power  —  if  such  a  term  may  properly  be 
used  in  connection  with  materials  —  lies  in  their  elasticity.  If  that 
is  damaged  the  life  of  the  material  is  damaged,  and  its  powers 
of  resistance  "are  weakened.  As  we  have  seen  in  the  preceding 
pages,  there  is  no  known  relation  between  the  coefficient  of  elas- 
ticity, and  the  ultimate  strength  of  materials.  The  coefficient  of 
elasticity  may  be  high  and  the  modulus  of  strength  comparatively 
low.  In  other  words,  the  limit  of  elasticity  of  some  metals  may 
be  passed  by  a  strain  of  less  than  one-third  their  ultimate 
strength,  while  in  others  it  may  exceed  one-half  their  ultimate 
strength.  We  see,  then,  the  unphilosophical  mode  of  fixing  an 
arbitrary  modulus  of  safety,  or  even  a  factor  of  safety,  when 
they  are  made  in  reference  to  the  ultimate  strength.  But  an 
examination  of  the  results  of  experiments  shows  that  the  limit 
of  elasticity  is  rarely  passed  for  strains  which  are  less  than  one- 
third  of  the  ultimate  strength  of  the  metal,  and  hence,  according 

*  Civ.  Eng.  and  Arch.  Jour.,  Vol.  xxiv.,  p.  329. 
f  Civ.  Eng.  and  Arch.  Jour.  ,  Vol.  xxiv.  ,  p.  226. 


LIMITS  OF  SAFE  LOADING  OF  MECHANICAL  STRUCTURES.        223 

to  the  views  of  the  engineers  given  in  the  preceding  article, 
the  factor  of  safety  is  generally  safe.  But  if  the  limit  of  elas- 
ticity were  definitely  known  it  is  quite  possible  that  a  smaller 
factor  of  safety  might  sometimes  be  used.* 

This  method  of  determining  the  limit  has  been  recognized  by 
some  writers,  and  the  propriety  of  it  has  been  admitted  by  many 
practical  men,  but  the  difficulty  of  determining  the  elastic  limit 
has  generally  precluded  its  use.  The  experiments  which  are 
necessary  for  determining  it  are  necessarily  more  delicate  than 
those  for  determining  the  ultimate  strength. 

There  is  also  a  slight  theoretical  objection  to  its  use.  The 
limit  of  elasticity  is  not  a  definite  quantity,  for  it  is  not  pos- 
sible to  determine  the  exact  point  where  the  material  is  over- 
strained. But  this  is  not  a  fatal  objection,  for  the  limit  can  be 
determined  within  small  limits. 

In  regard  to  the  margin  that  should  be  left  for  safety,  much 
depends  upon  the  character  of  the  loading.  f  If  the  load  is 
simply  a  dead  weight,  the  margin  may  be  comparatively  small ; 
but  if  the  structure  is  to  be  subjected  to  percussive  forces  or 
shocks,  it  is  evident,  as  indicated  in  articles  19  and  171,  that 
the  margin  should  be  comparatively  large,  not  only  on  account 
of  the  indeterminate  effect  of  the  force,  but  also  on  account  of 
the  effect  of  such  a  force  upon  the  resisting  powers  of  the 
material.  In  the  case  of  railroad  bridges,  for  instance,  the 
vertical  posts  or  ties,  as  the  case  may  be,  are  generally  subjected 
to  more  sudden  strains  due  to  a  passing  load,  than  the  upper 
and  lower  chords,  and  hence  should  be  relatively  stronger. 
The  same  remark  applies  to  the  inclined  ties  and  braces  which 
form  the  trussing;  and  to  any  parts  which  are  subjected  to 
severe  local  strains. 

The  frames  of  certain  machines,  and  parts  of  the  same 
machines,  are  subjected  to  a  constant  jar  while  in  use,  in  which 
cases  it  is  very  difficult  to  determine  the  proper  margin  which 
is  consistent  with  economy  and  safety.  Indeed,  in  such  cases, 
economy  as  well  as  safety  generally  consists  in  making  them 

*  James  B.  Eads,  in  his  Report  upon  the  Illinois  and  St.  Louis  Bridge,  for 
1871,  states  that  he  tested  samples  of  steel  which  were  to  be  used  in  that 
structure,  which  showed  limits  of  elastic  reaction  of  70,000  to  93,000  pounds 
per  square  inch. 


224  THE   RESISTANCE    OF   MATERIALS. 

excessively  strong,  as  a  single  breakage  might  cost  much  more 
than  the  extra  material  necessary  to  fully  insure  safety. 

The  mechanical  execution  of  a  structure  should  be  taken 
into  consideration  in  determining  the  proper  value  of  the  mar 
gin  of  safety.  If  the  joints  are  imperfectly  made,  excessive 
strains  may  fall  upon  certain  points,  and  to  insure  safety,  the 
margin  should  be  larger.  No  workmanship  is  perfect,  but  the 
elasticity  of  materials  is  favorable  to  such  imperfections  as 
necessarily  exist ;  for,  when  only  a  portion  of  the  surface  which 
is  intended  to  resist  a  strain,  is  brought  into  action,  that  por- 
tion is  extended  or  compressed,  as  the  case  may  be,  and  thus 
brings  into  action  a  still  larger  surface.  But  workmanship 
which  is  so  badly  executed  as  to  be  considered  imperfect 
would  fail  before  all  its  parts  could  be  brought  into  bearing. 

170.     EXAMPLES    OF  STRAINS  THAT    HAVE  BEEN    USED 

IN  PRACTICAL  CASES.  The  margin  of  safety  that  has  been 
used  in  various  structures  may  or  may  not  serve  as  guides 
in  designing  new  structures.  If  the  margin  for  safety  is  so 
small  that  the  structure  appears  to  be  insecure  and  gives  indi- 
cations of  failure,  it  evidently  should  not  be  followed.  It 
serves  as  a  warning  rather  than  as  a  guide.  If  the  margin 
is  evidently  excessively  large,  demanding  several  times  the 
amount  of  material  that  is  necessary  for  stability,  it  is  not  a 
guide.  Any  engineer  or  mechanic,  without  regard  to  scientific 
skill  or  economy  in  the  use  of  materials,  may  err  in  this  direc- 
tion to  any  extent.  But  if  the  margin  appears  reasonably  safe, 
and  the  structure  has  remained  stable  for  a  long  time,  it  serves 
as  a  valuable  guide,  and  one  which  may  safely  be  followed 
under  similar  circumstances.  Structures  of  this  kind  are 
practical  cases  of  the  approximate  values  of  the  inferior  limits 
of  the  factors  of  safety.  The  following  are  some  practical 
examples : — 


LIMITS  OF  SAFE  LOADING  OF  MECHANICAL  STRUCTURES. 


225 


IRON   TRUSSED   BRIDGES. 


NAME  OP  THE  BRIDGE. 

TENSION. 

Tons 
per  square  inch. 

COMPRESSION. 

Tons 
per  square  inch. 

Passaic  (Lattice)  

5*  to  6 
4 
5 
5 
5 
5 

Pounds 
per  square  inch. 

12,000 
7,000  to  12,000 
9,251 
10,000 
11,375 
Factor  of  safety,  5 

4i  to  5± 

^f 

5 

4 

Pounds 
per  square  inch. 

,12,000 
i  to  \  the  strength 
8,902 
Factor  of  safety,  5 
711 
Factor  of  safety,  5 

Place  de  FEurope  (Lattice)   

Canastota  (N  Y  C  R.  It  )  (Lattice) 

Newark  Dyke  (  Warren  Girder)     

Charring  Cross  (Lattice) 

St  Charles  Mo.  (  Whipple  Truss)  *     .... 

Louisville,  Ky.  (Fink  Truss)  

Quincy  Bridge  ^                     .... 

Kansas  City  Bridge  §         

Hannibal  Bridge  (Quadrangular  Truss)  \. 

WOODEN   BRIDGES. 


NAME  OF  THE  BRIDGE. 


MAXIMUM  STRAIN. 


Cumberland  Valley  R.  R.  Bridge 

Portage  Bridge  (N.  T.  &  E.  R.  R.). 


635  pounds  per  square  inch. 
Factor  of  safety  20. 


*  R  R  Gazette,  July  8,  1871,  p.  169. 

f  R  R  Gazette,  July  15,  1871,  p.  178.     Pivot  span  376  feet  5  inches ;  longest 
pivot  span  yet  constructed. 

\  Report  of  Chief  Engineer  Clark. 

§  Calculated  from  the  Report  of  Chief  Engineer  0.  Chanute,  pp.  106  and 
136. 

||  The  tensile  strength  of  the  material  ranged  from  55,000  Ibs.  to  65,000 
Ibs.  per  square  inch.—  R  R  Gazette,  July  15,  1871,  p.  169. 
15 


226 


THE   RESISTANCE    OF   MATERIALS. 


CAST-IRON   ARCHES.* 


NAME  OF  THE  AKCH. 

SPAN. 

Feet.     Inches. 

VERSED  SINE. 

Feet.    Inches. 

STRAIN 
PEK  SQUARE 
INCH  IN  TONS. 

Austerlitz 

106 
152 
102 
137 
197 
120 

0 
2 
5 

9 
10 
0 

10 
16 
11 
15 
16 
20 

7 
1 
4 
0 
5 
0 

2.78 
1.46 
1.37 
1.90 
2.87 
3.00 

Carrousal     .... 

St.  Denis  

Nevers  

Rhone  

Westminster 

STONE   ARCHES,  f 


| 

>-.<*  s 

•g 

a 

^  ^ 

NAME  OF  THE  ARCH. 

1 
c 

1 

pll 

"o  c  o 

s 

»  §  §  >> 

fe^g 

9 

g 

s  nT  a,s 

ftss 

t 

02 

t> 

* 

fe 

100 

15 

175- 

11  3 

Waterloo  (9  Arches)  .  .  . 

120 

35 

151 

20.0 

Neuilly      

128 

32 

172 

11.6 

Taaf  (South  Wales)  

140 

35 

244 

8.0 

Turin  

147 

18 

293 

10.2 

152 

38 

215 

14.0 

Chester                    .... 

200 

42 

349 

8.6 

CAST-STEEL  ARCH. 


NAME  OF  ARCH. 

SPAN. 

Feet. 

FACTOR 
OF  SAFETY. 

515 

6  +  t 

*  Irwin  on  Iron  Bridges  and  Roofs. 

f  Cresy's  Encyclopsedia. 

\  Report  of  the  Engineer,  p.  33. 


LIMITS  OF  SAFE  LOADING  OF  MECHANICAL  STRUCTURES.        227 


SUSPENSION   BRIDGES. 


4 

iit 

41 

• 

J 

£-££ 

.8-Sh3 

«s 

B   B'C 

c  g-o 

1 

NAME  OF  THE  BRIDGE. 

a 

C  P 

ill 

•3 

I 

2  «g 

•g  cr  4> 

| 

8g£ 

m  En 

I 

Menai      ... 

580 

4  21 

8  00 

3  9* 

Hammersmith  

5  38 

9  36 

3  3* 

Pesth  

666 

5.01 

8  11 

3.9* 

Chelsea          .     . 

384 

4.36 

8  07 

3  9* 

Clifton  

702J 

2  90 

5  03 

6  4* 

Niagara  

821 

6.70 

8  40 

5  3+ 

Suspension  Aqueduct,   Pitts-  [ 
burgh,  Pa.     7  spans,  each,  ) 

160 

4.0 

Cincinnati  Bridge  ||  

1,057 

9.1 

11  7      - 

6  2 

East  River  

1,600:}: 

6.0 

Highland  (proposed) 

6.0 

TUBULAK    BRIDGES. 


SPAN. 

FOB  WEIGH 
AND 

r  OF  BRIDGE 
LOAD. 

NAME  OF  BRIDGE. 

Feet. 

Tension. 
Tons. 

Compression. 
Tons. 

400 

6.85 

5.03 

460 

3.00 

4.75 

4.25 

*  Tensile  strength,  70,000  Ibs.  per  square  inch. 

f  Tensile  strength,  100,000  Ibs.  per  square  inch, 

\  Engineer's  Report.     Suspending  ties,  factor  of  safety,  8. 

§  Jour.  Frank.  Inst.,  vol  LXXXVII.,  p.  165. 

\   Report  of  the  Chief  Engineer,  J.  A.  Roebling. 


228  THE   RESISTANCE    OF   MATERIALS. 


6TO1?E    FOUNDATIONS. 


FACTOR  OF  SAFETY. 

Pillars  of  the  Dome  of  St.  Peter's  (Rome)  

16 

u         "                "St.  Paul's  (London)  

14 

'  '     St.  Geneviere  (Paris) 

7  6 

Pillars  of  the  Church  Touissaint  (Angers)  

10 

Merchants'  Shot  Tower  (Baltimwe)*  

4  8 

Lower  courses  of  Britannia  Bridge  

31 

Lower  courses  of  the  Piers  of  Neuilly  Bridge  (Paris)  

15  8 

Foundation  of  St.  Charles'  Bridge  (Missouri]   .  . 

12  to  14 

Foundations  of  East  River  Bridgef  

10  to  20 

177.  PROOF  LOAD.  The  proof  load  is  a  trial  load.  It 
is  intended  as  a  practical  test  of  a  theoretical  structure. 

It  generally  exceeds  the  greatest  load  that  it  is  ever  intended 
to  put  upon  the  structure.  Some  structures,  especially  steam 
boilers,  are  subjected  to  excessively  severe  trial  strains,  often 
being  two  or  even  three  times  that  to  which  they  are  to  be 
subjected  in  actual  use.  "  This  is  done,"  they  say,  "  to  insure 
against  failure."  But  such  severe  strains  do  no  good,  and 
often  do  great  damage,  for  they  may  overstrain  the  material 
and  thus  weaken  it.  For  instance,  if  a  structure  is  proportioned 
to  carry  10,000  pounds  per  square  inch,  but  on  account  of  the 
imperfection  of  the  material  or  imperfect  workmanship,  some 
point  is  required  to  carry  20,000  pounds  per  square  inch ;  and  if  the 
structure  is  tested  to  20,000  pounds  per  square  inch,  that  point 
would  be  obliged  to  carry  nearly  40,000  pounds  per  square  inch,  an 
amount  which  it  might  not  be  able  to  sustain  for  a  long  time. 
We  say  nearly  40,000  pounds ;  for  it  should  be  remembered  that 
where  the  workmanship  is  imperfect,  thereby  throwing  greater 
strains  on  certain  points  than  was  intended,  the  elasticity  of 
the  materials  permits  those  members  to  be  compressed  or 
extended  more  than  they  otherwise  would  be,  and  thus  tends 

*  Strength  of  materials,  J.  K.  Whildin,  p.  23. 

f  "In  the  stone  work  the  pressures  vary  from  8  to  26  tons  per  square  foot. 
Stone  used  is  granite,  selected  samples  of  which  have  borne  a  crushing  strain 
of  600  tons  per  square  foot.  Some  will  not  bear  over  100  tons  per  square  foot. 
The  general  average  is  necessarily  much  less  than  that  of  the  best  specimens. " 
— Statement  of  the  Chief  Engineer,  Washington  A.  Roebling. 


LIMITS  OF  SAFE  LOADING  OF  MECHANICAL  STRUCTURES.        229 

to  bring  into  action  the  other  members  which  at  first  were 
more  lightly  strained.  In  this  way  there  is  a  tendency  to  bring 
about  an  equilibrium  of  strains  on  all  those  parts  which  were 
calculated  to  carry  equal  amounts. 

According  to  the  *  principles  which  have  been  discussed 
in  the  preceding  pages,  it  is  evidently  better  for  the  struc- 
ture, and  should  be  more  satisfactory,  to  apply  a  moderate 
proof  load  for  a  long  time  than  an  excessive  one  for  a  short 
time. 


APPENDIX  I. 


PRESERVATION  OF  TIMBER. 

(The  Graduating  Thesis  of  Mr.  H.  W.  Lewis,  of  the  class  of  1866,  and  more 
recently  engineer  on  the  Missouri  Valley  Railroad,  forms  the  basis  of  this 
article.  I  have  added  to  it  such  new  matter  as  I  find  in  the  Graduating  Thesis 
of  Mr.  A.  B.  Raymond,  of  the  class  of  1871. — AUTHOR.) 

1.  THE  IMPORTANCE  OF  THIS  SUBJECT  maybe  shown  by  many  familiar 
examples  in  practical  life. 

Although  iron  is  coming  more  and  more  extensively  into  use,  yet  the  amount 
of  wood  which  is  used  at  the  present  time  in  mechanical  structures,  and  which 
will,  in  the  nature  of  things,  be  used  for  a  long'  time  to  come,  is  enormous. 
For  instance:  in  1865  there  was  sold  in  Chicago  alone  900,000,000  pieces  of 
lath,  2,000,000,000  of  shingles,  and  5,000,000,000  feet  of  lumber.* 

In  the  matter  of  railroad  ties  alone,  any  process  which  could  be  easily  and 
cheaply  applied,  which  would  double  their  life,  would  literally  save  millions  to 
the  country.  This  may  be  shown  by  an  approximate  calculation,  thus :  — Al- 
lowing only  2,000  sleepers  to  the  mile,  at  a  cost  of  fifty  cents  each,  and  admit- 
ting that  the  average  life  of  American  sleepers  is  only  seven  years,f  and  that 
it  costs  ten  cents  to  treat  each  tie  in  some  way  so  as  to  make  it  last  fourteen 
years,  then  the  saving  at  the  end  of  seven  years  is  $600  per  mile.  For  ten 
cents  at  compound  interest  at  ten  per  cent,  for  seven  years  amounts  to  twenty 
cents,  which  from  fifty  cents  leaves  thirty  cents  as  the  net  saving  on  one ;  and 
on  2,000  it  amounts  to  $600. 

There  are  in  the  United  States  about  45,000  miles  of  railroad;  and  hence, 
if  the  above  conditions  could  be  realized  of  all  of  them,  the  annual  saving 
would  be  about  $3,400,000  !  Other  uses  of  timber  would  show  a  correspond- 
ing saving. 

2.  CLASSIFICATION  OF  CONDITIONS.— Timber  may  be  subjected  to  the 
following  conditions :  — 

It  may  be  kept  constantly  dry  ;  at  least,  practically. 

It  may  be  constantly  wet  in  fresh  water. 

It  may  be  constantly  damp. 

It  may  be  alternately  wet  and  dry. 

It  may  be  constantly  wet  in  sea  water. 

3.  TIMBER  KEPT  CONSTANTLY  DRY  will  last  for  centuries.  The  roof 
of  Westminster  Hall  is  more  than  450  years  old.  In  Stirling  Castle  are  carv- 
ings in  oak,  well  preserved,  over  300  years  old ;  and  the  trusses  of  the  roof  of 


*  Hunt's  Merchants'1  Magazine. 

t  New  American  Cyclopedia,  vol.  xiii.,  p.  734. 


232  APPENDIX. 

the  Basilica  of  St.  Paul,  Rome,  were  sound  and  good  after  1,000  years  of 
service.*  The  timber  dome  of  St.  Mark,  at  Venice,  was  in  good  condition 
850  years  after  it  was  built,  f 

Artificial  preservatives  seem  to  be  unnecessary  under  this  condition. 

4.  TIMBER  KEPT  CONSTANTLY  WET  IN   FRESH  WATER,  under   such 
conditions  as  to  exclude  the  air,  is  also  very  durable.     The  pillars  upon  which 
dwellings  of  the  Canaries  rest  were  put  in  their  present  place  in  1402,  and 
they  remain  sound  to  the  present  time  4     The  utensils  of  the  lake  dwellings 
of  Switzerland  are  supposed  to  be  at  least  2,000  years  old.§ 

The  piles  of  the  old  London  Bridge  were  sound  800  years  after  they  were 
driven.  The  piles  of  bridge  built  by  Trajan,  after  having  been  driven  more 
than  1,600  years,  were  found  to  be  petrified  four  inches,  the  rest  of  the  wood 
being  in  its  ordinary  condition.  § 

Beneath  the  foundation  of  Savoy  Place,  London,  oak,  elm,  beech,  and 
chestnut  piles  and  planks  were  found  in  a  perfect  state  of  preservation  after 
having  been  there  650  years.  § 

While  removing  the  old  walls  of  Tunbridge  Castle,  Kent,  there  was  found, 
in  the  middle  of  a  thick  stone  wall,  a  timber  curb  which  had  been  enclosed 
for  700  years.  § 

It  is  doubtful  if  artificial  preparations  would  have  prolonged  the  life  of  the 
timber  in  these  cases. 

5.  TIMBER  IN  DAMP  SITUATIONS.— Timber,   in  its  native   state,   under 
these  circumstances,  is  liable  to  decay  rapidly  from  the  disease  called  "dry 
rot."     In  dry  rot  the  germs  of  the  fungi  are  easily  carried  in  all  directions  in 
a  structure  where  it  has  made  its  appearance,  without  actual  contact  between 
the  sound  and  decayed  wood  being  necessary ;  whereas  the  communication  of 
the  disease  resulting  from  wet  rot  takes  place  only  by  actual  contact.     The 
fungus  is  not  the  cause  of  the  decay,  but  only  converts  corrupt  matter  into 
new  forms  of  life.  || 

There  are  three  conditions  which  are  at  our  command  for  prolonging  the  life 
of  timber  in  damp  situations  : — 

1st.  Thoroughly  season  it ; 

2d.  Keep  a  constant  circulation  of  air  about  it ;  and 

3d.  Cover  it  with  paint,  varnish,  or  pitch. 

The  first  condition  is  essential,  and  may  be  combined  with  either  or  both  of 
the  others. 

By  seasoning  we  do  not  mean  simply  drying  so  as  to  expel  the  water  of  the 
sap,  but  also  a  removal  or  change  of  the  albuminous  substances.  These  are 
fermentable  substances,  and  when  both  are  present  they  are  ever  ready,  under 
suitable  circumstances,  to  promote  decay.  The  cellulose  matter  of  the  woody 
fibre  is  very  durable  when  not  acted  upon  by  fermentation,  and  it  is  this  that 
we  desire  especially  to  protect. 


*  The  London  Builder,  vol.  ii.,  p.  616. 

t  Modern  Carpentry,  Silloway,  p.  40. 

t  Journal  of  the  Frank.  Inst.,  1870. 

§  Modern  Carpentry,  Silloway,  p.  39. 

||  ' '  There  is  no  reason  to  believe  that  fungi  can  make  use  of  organic  compounds  in  any  other  than 
a  state  of  decomposition." — Carpenter's  Comp.  Physiology,  p.  165.  (See  also  Encyclopaedia  Bri- 
tanica  on  this  subject.) 


APPENDIX.  233 

Unseasoned  timber  which  is  surrounded  by  a  dead  air  decays  very  rapidly. 
The  timber  of  many  modern  constructions  is  translated  from  the  forests  and 
enclosed  in  a  finished  building  in  a  few  weeks,  and  unless  it  is  subject  to  a  free 
circulation  of  air  it  inevitably  decays  rapidly.  * 

Thorough  ventilation  is  indispensable  to  the  preservation  of  even  well-sea- 
soned naked  wood  in  damp  localities.  The  rapid  decomposition  of  sills,  sleep- 
ers, and  lower  floors  is  not  surprising  where  neither  wall-gratings  nor  venti- 
lating flues  carry  off  the  moisture  rising  from  the  earth,  or  foul  gases  evolved 
in  the  decay  of  the  surface  mould.  In  the  close  air  of  cellars,  and  beneath 
buildings,  the  experiments  of  Pasteur  detected  the  largest  percentage  of 
fungi  spores.  .  Remove  the  earth  to  the  foot  of  the  foundation,  and  fill  in  the 
cavity  with  dry  sand,  plaster-rubbish,  etc. ,  or  lay  down  a  thick  stratum  of 
cement  to  exclude  the  water,  and  provide  for  a  complete  circulation  of  air, 
and  lower  floors  will  last  nearly  as  long  as  upper  ones.f 

A  covering  of  paint,  pitch,  varnish,  or  other  impervious  substance  upon  un- 
dried  timber  is  very  detrimental,  for  by  it  all  the  elements  of  decay  are  re- 
tained and  compelled  to  do  their  destroying  work.  The  folly  of  oiling,  paint- 
ing, or  charring  the  surface  of  unseasoned  wood  is  therefore  evident.  Owing 
to  this  blunder  alone,  it  is  no  unusual  thing  to  find  the  painted  wood-work  of 
older  buildings  completely  rotted  away,  while  the  contiguous  naked  parts  are 
perfectly  sound. 

While  an  external  application  of  coal  tar  promotes  the  preservation  of  dry 
timber,  nothing  can  more  rapidly  hasten  decay  than  such  a  coating  upon  the 
surface  of  green  wood.  But  this  mistake  is  often  made,  and  dry  rot  does  the 
work  of  destruction.  \  Carbonizing  the  surface  also  increases  the  durability  of 
dry,  but  promotes  the  decay  of  wet  timber.  Farmers  very  often  resort  to 
one  of  the  latter  methods  for  the  preservation  of  their  fence-posts.  Unless 
they  discriminate  between  green  and  seasoned  timber,  these  operations  will 
prove  injurious  instead  of  beneficial. 

There  are  numerous  methods  for  promoting  the  process  of  seasoning. 
Some  have  in  view  simply  drying,  a  process  which  is  important  in  itself,  but 
which  will  not  in  itself  prevent  decay  in  damp  situations  unless  the  moisture 
be  permanently  excluded.  Some  dry  with  hot  air,  and  some  with  steam.  In 
the  latter  case,  if  the  steam  be  superheated  the  process  is  very  rapid,  but  it 
seems  to  damage  the  life  of  the  timber. 

Others  have  in  view  the  expulsion  of  the  albuminous  substance.  Water-soak- 
ing the  logs  and  afterwards  drying  the  lumber,  seems  to  be  a  cheap  and  quite 
effectual  mode.  But  there  are  many  patented  processes  for  securing  this  end, 
or  for  changing  the  albuminous  substances ;  and  in  many  cases  the  latter  end 
is  not  only  secured,  but  the  salts  which  are  used  act  directly  upon  the  cellulose 
and  lignite  of  the  wood,  thereby  greatly  promoting  its  durability. 


*  For  an  account  of  the  rapid  destruction  of  the  floors  and  joists  of  the  Church  of  the  Holy 
Trinity,  Cork,  Ireland,  by  dry  rot,  see  Civil  Engineer's  Journal,  vol.  xii.,  p.  303.  For  an  ac- 
count of  the  decay  of  floors,  studs,  &c.,  in  a  dwelling,  see  the  London  Builder,  vol.  vi.,  p.  34. 

"In  some  of  the  mines  in  France  the  props  seldom  last  more  than  fifteen  months." — Annalen 
des  Mines. 

t  The  Builder,  vol.  xi.,  page  46. 

%  According  to  Col.  Berrien,  the  Michigan  Central  Railroad  bridge,  at  Niles,  was  painted, 
before  seasoning,  with  "  Ohio  fire-proof  paint,"  forming  a  glazed  surface.  About  five  years  after, 
it  was  so  badly  dry-rotted  as  to  require  rebuilding. 


234  APPENDIX. 

The  following  are  the  principal  processes  which  have  been  used :  — Mr.  P. 
W.  Barlow's  patent*  provided  for  exhausting  the  air  from  one  end  of  the  log, 
while  one  or  more  atmospheres  press  upon  the  other  end.  This  artificial 
aerial  circulation  through  the  wood  is  prolonged  at  pleasure.  However  excel- 
lent in  theory,  this  process  is  not  practicable. 

By  another  method,  the  smoke  and  hot  gases  of  a  coal  fire  are  conveyed 
among  the  lumber,  placed  in  a  strong  draft.  Some  writers  recommend  the  re- 
moval of  the  bark  one  season  before  felling  the  tree.  All  good  authorities 
agree  that  the  cutting  should  take  place  in  the  winter  season,  f 

Kyan's  process,  which  consists  in  the  use  of  corrosive  sublimate,  was  pat- 
ented in  1832.  His  specific  solution  |  was  one  pound  of  chloride  of  mercury 
to  four  gallons  of  water.  Long  immersion  in  the  liquid  in  open  vats,  or  great 
pressure  upon  both  solution  and  wood,  in  large  wrought-iron  tanks,  is  neces- 
sary for  the  complete  injection  of  the  liquid.  The  durability  of  well  kyanized 
timber  has  been  proved,  but  the  expensiveness  of  the  operation  will  long  for- 
bid its  extensive  adoption. 

For  "  Burnettizing,"  §  a  solution  of  chloride  of  zinc — one  pound  of  salt  to 
ten  gallons  of  water — is  forced  into  the  wood  under  a  pressure  of  150  Ibs.  per 
square  inch. 

Boucherie  employs  a  solution  ||  of  sulphate  of  copper  one  pound  to  water 
twelve  and  a  half  gallons,  or  pyrolignite  of  iron  one  gallon  to  six  gallons  of  water. 
He  enclosed  one  end  of  the  green  stick  in  a  close-fitting  collar,  to  which  is  at- 
tached an  impervious  bag  communicating  through  a  flexible  tube  with  an 
elevated  reservoir  containing  the  salt  liquid.  Hydrostatic  pressure  soon  expels 
the  sap  at  the  opposite  end  of  the  log.  When  the  solution  makes  its  appear- 
ance also  the  process  is  completed. 

He  finds  the  fluid  will  pass  along  the  grain,  a  distance  of  12 'feet,  under  a 
lower  pressure  than  is  required  to  force  it  across  the  grain,  three-fourths  of  an 
inch.  The  operation  is  performed  upon  green  timber  with  the  greatest  fa- 
cility. 1 

In  1846,  eighty  thousand  sleepers  of  the  most  perishable  woods,  impregnated, 
by  Boucherie's  process,  with  sulphate  of  copper,  were  laid  down  on  French 
railways.  After  nine  years'  exposure,  they  were  found  as  perfect  as  when  laid.** 
This  experiment  was  so  satisfactory  that  most  of  the  railways  of  that  empire 
at  once  adopted  the  system.  We  would  suggest  washing  out  the  sap  with 
water,  which  would  not  coagulate  its  albumen.  The  solution  would  appropri- 
ately follow. 

Both  of  the  last-named  processes  are  comparatively  cheap.  The  manufac- 
turing companies  of  Lowell,  Massachusetts,  have  an  establishment  for  "Bur- 
nettizing" timber,  ff  in  which  they  prepare  sticks  fifty  feet  in  length.  Under 

*  Civ.  Eng.  Jour.,  vol.  xix.,  p.  422. 

t  Experiments  detailed  in  the  Cosmos  show  conclusively  that  winter-cut  pine  is  stronger  and 
more  durable  than  that  cut  at  any  other  season  of  the  year. — Ann.  Sc.  Discovery  for  1861,  p.  346. 

"Oak  trees  felled  in  the  winter  make  the  best  timber."— The  Builder,  1859,  page  138. 

%  Civ.  Eng.  Jour.,  vol.  v.,  page  202. 

§  Civ.  Eng.  Jour.,  vol.  xiv.,  p.  471.     Invented  by  Burnett  in  1838. 

H  Civ.  Eng.  Jour.,  vol.  xx.,  p.  405. 

1  As  a  modification  of  this  method  he  also  cut  a  channel  in  the  wood  throughout  the  circumfer- 
ence of  the  tree,  fitted  a  reservoir  thereunto,  and  poured  in  the  liquid.  The  vital  forces  speedily 
disseminated  the  solution  throughout  the  tree. 

**  Jour,  of  the  Frank.  Inst.,  vol.  xxxii.,  pp.  2,  3. 

tt  New  American  Cyclopaedia. 


APPENDIX.  235 

a  pressure  of  125  pounds  per  square  inch  they  inject  from  two  to  eight  ounces 
of  the  salt  into  each  cubic  foot  of  wood.  The  cost,  in  1861,  was  from  $5  to. 
$6  per  1,000  feet,  board  measure.*  Boucherie's  method  must  be  still  cheaper. 
It  costs  less  than  creosoting  by  one  shilling  per  sleeper,  f 

An  American  engineer,  Mr.  Hewson,  for  injecting  railroad  sleepers,  proposes 
a  vat  deep  enough  for  the  timbers  to  stand  in  upright.  The  pressure  of  the 
surrounding  solution  upon  the  Idwer  ends  of  the  sticks  will,  he  thinks,  force 
the  air  out  at  their  upper  extremities,  kept  just  above  the  surface  of  the 
solution,  after  which  the  latter  will  rise  and  impregnate  the  wood.  In  1859 
he  estimated  chloride  of  zinc  at  9  cents  per  pound,  sulphate  of  copper  at  14 
cents  per  pound,  and  pyrolignite  of  iron  at  23  cents  per  gallon.  He  found  the 
cost  of  impregnating  a  railway  tie  with  sufficient  of  those  salts  to  prevent 
decay,  to  be :  for  the  chloride  of  zinc  2 '8  cents,  for  blue  vitriol  3*24  cents,  for 
pyrolignite  of  iron  7*5  cents.:}: 

By  Eaiie's  process  the  timber  is  boiled  in  a  solution  of  one  part  of  sulphate 
of  copper,  three  parts  of  the  sulphate  of  iron,  and  one  gallon  of  water  to 
every  pound  of  the  salts.  A  hole  was  bored  the  whole  length  of  the  piece 
before  it  was  boiled.  It  was  boiled  from  two  to  four  hours,  and  allowed  to  cool 
in  the  mixture. 

Ringold  and  Earle  invented  the  following  process  :  — A  hole  was  made  the 
whole  length  of  the  piece,  from  one-half  to  two  inches  in  diameter,  and  boiled 
from  two  to  four  hours  in  lime-water.  After  the  piece  was  dried  the  hole  was 
filled  with  lime  and  coal  tar.  Neither  of  these  methods  was  very  successful. 

A  Mr.  Darwin  suggests  that  the  piece  be  soaked  in  lime-water,  and  after- 
wards in  sulphuric  acid,  so  as  to  form  gypsum  in  the  pores. 

Bethell's  process  consists  in  forcing  dead  oil  into  the  timber.  This  is  called 
creosoting.  §  He  inclosed  the  timber  and  dead  oil  in  huge  iron  tanks,  and  sub- 
jected them  to  a  pressure  varying  between  100  and  200  pounds  per  square 
inch,  at  a  temperature  of  120°  F.  about  twelve  hours.  From  eight  to  twelve 
pounds  of  oil  are  thus  injected  into  each  cubic  foot  of  wood.  Lumber  thus- 
prepared  is  not  affected  by  exposure  to  air  and  water,  and  requires  no  painting.  || 
A  large  number  of  English  railway  companies  have  already  adopted  the 
system.^"  Eight  pounds  of  oil  per  cubic  foot  is  sufficient  for  railway  sleepers.** 

One  writer  has  said  that  if  creosote  has  ever  failed  to  prevent  decay,  it  has 
been  because  of  an  improper  treatment,  or  because  the  oil  was  deficient  in  car- 
bolic acid. 

Sir  Robert  Smirke  was  one  of  the  first  architects  to  use  this  process,  and 
when  examined  before  a  Committee  on  Timber,  stated  that  this  process  does  not 


*  The  Philadelphia,  Wilmington  and  Baltimore  Railroad  Company  have  used  the  process  since 
I860  with  complete  success.  The  Union  Pacific  Railroad  Company  have  recently  erected  a  large 
building  for  this  purpose.  Their  cylinder  is  75  feet  long,  61  inches  in  diameter,  and  capable  of 
holding  250  ties.  They  "  Burnettize"  two  batches  per  day.— Report  on  Pacific  Railroad,  by  Col. 
Simpson,  1865. 

t  Jour.  Frank."  Inst.,  vol.  xxxii.,  pp.  2,  3. 

\  Ibid.,  vol.  xxvii.,  p.  8. 

§"  Creosote  from  coal  undoubtedly  contains  two  homologous  bodies,  CiaHoOa  and  CuHsOa, 
the  first  being  carbolic  and  the  second  crysilic  acid."— Ure's  Diet,  of  Arts,  Manu.,  and  Mines,  vol. 
ii.,  p.  623. 

|  Ure's  Diet,  of  Manu.  and  Mines. 

T  The  Great  Western,  North-Eastern,  Bristol  and  Exeter,  Stockton  and  Darlington,  Manchester 
and  Birmingham,  aad  London  and  Birmingham. — Ure's  Diet,  of  Manu.  and  Mines. 

**  Jour.  Frank.  Inst.,  vol.  xliv.,  p.  275. 


236  APPENDIX. 

diminish  the  strength  of  the  material  which  is  operated  upon.  He  afterwards 
said,  "  I  cannot  rot  creosoted  timber,  and  I  have  put  it  to  the  severest  test  I 
could  apply." 

The  odor  of  creosote  makes  it  objectionable  for  residences  and  public 
buildings. 

Mr.  S.  Beer,  of  New  York  City,  invented  a  mode  of  preserving  timber  by 
boiling  it  in  borax  with  water.  But  this  process  has  been  objected  to  on  the 
ground  that  it  is  not  a  good  protection  against  moisture. 

Common  salt  is  known  to  be  a  good  preservative  in  many  cases.  According 
to  Mr.  Bates' s  opinion,*  it  answers  a  good  purpose  in  many  cases  if  the  pieces 
to  which  it  is  exposed  are  not  too  large. 

6.  TIMBER  AI/TERNATEL.Y  WET  AND  DRY.— The  surface  of  all  timber 
exposed  to  alternations  of  wetness  and  dryness,  gradually  wastes  away,  be- 
coming dark-colored  or  black.  This  is  really  a  slow  combustion,  but  is  com- 
monly called  wet  rot,  or  simply  rot.  Other  conditions  being  the  same,  the 
most  dense  and  resinous  woods  longest  resist  decomposition.  Hence  the  su- 
perior durability  of  the  heart-wood,  in  which  the  pores  have  been  partly  filled 
with  lignine,  over  the  open  sap-wood,  and  of  dense  oak  and  lignum-vitse 
over  light  poplar  and  willow.  Hence,  too,  the  longer  preservation  of  the 
pitch-pine  and  resinous  "  jarrah  "  of  the  East,  as  compared  with  non- resinous 
beech  and  ash. 

Density  and  resinousness  exclude  water.  Therefore  our  preservatives 
should  increase  those  qualities  in  the  timber.  Fixed  oils  fill  up  the  pores  and 
increase  the  density.  Staves  from  oil-barrels  and  timbers  from  whaling  ships 
are  very  durable.  The  essential  oils  resinify,  and  furnish  an  impermeable  coat- 
ing. But  pitch  or  dead  oil  possesses  advantages  over  all  known  substances  for 
the  protection  of  wood  against  changes  of  humidity.  According  to  Professor 
Letheby,f  dead  oil,  1st,  coagulates  albuminous  substances  ;  2d,  absorbs  and 
appropriates  the  oxygen  in  the  pores,  and  so  protects  from  eremacausis  ;  3d, 
resinifies  in  the  pores  of  the  wood,  and  thus  shuts  out  both  air  and  moisture  ; 
and  4th,  acts  as  a  poison  to  lower  forms  of  animal  and  vegetable  life,  and  so  pro- 
tects the  wood  from  all  parasites.  All  these  properties  specially  fit  it  for  im- 
pregnating timber  exposed  to  alternations  of  wet  and  dry  states,  as,  indeed, 
some  of  them  do,  for  situations  damp  and  situations  constantly  wet.  Dead  oil 
is  distilled  from  coal-tar,  of  which  it  contains  about  .30,  and  boils  between 
390°  and  470°  Fahr.  Its  antiseptic  quality  resides  in  the  creosote  it  contains. 
One  of  the  components  of  the  latter,  carbolic  acid  (phenic  acid,  phenol), 
Ci2H602,  the  most  powerful  antiseptic  known,  is  able  at  once  to  arrest  the  de- 
cay of  every  kind  of  organic  matter 4  Prof.  Letheby  estimates  this  acid  at  -£ 


*  Report  of  the  Commissioner  of  Agriculture.  t  Civ.  Eng.  Jour.,  vol.  xxiii.,  p.  216. 

J  "  I  have  ascertained  that  adding  one  part  of  the  carbolic  acid  to  five  thousand  parts  of  a  strong 

solution  of  glue  will  keep  it  perfectly  sweet  for  at  least  two  years Hides  and 

skins,  immersed  in  a  solution  of  one  part  of  carbolic  acid  to  fifty  parts  of  water,  for  twenty-four 
hours,  dry  in  air  and  remain  quite  sweet.1'— Prof.  Grace  Calvert,  Ann.  Sc.  Discov.,  1865,  p.  55. 

"  Carbolic  acid  is  sufficiently  soluble  in  water  for  the  solution  to  possess  the  power  of  arresting 
or  preventing  spontaneous  fermentation .  Saturated  solutions  act  on  animals  and  plants  as  a  viru- 
lent poison,  though  containing  only  five  per  cent,  of  the  acid.1' — Civ.  Eng.  Jour. ,  vol.  xxii.,  p.  216. 

"  Parasites  and  other  worms  are  instantly  killed  by  a  solution  containing  only  one-half  per  cent. 

of  acid,  or  by  exposure  to  the  air  containing  a  small  portion  of  the  acid By 

examining  the  action  on  a  leaf,  we  find  the  albumen  is  coagulated.  All  animals  with  a  naked  skin, 
and  those  that  live  in  water,  die  sooner  than  those  that  live  in  air  and  have  a  solid  envelope." — 
Dr.  I.  Lemaire,  Ann.  Sc,  Discov. ,  1855,  p.  238. 


APPENDIX. 


237 


to  6  per  cent,  of  the  oil.  Chrysilic  acid  C14H,O3,  the  homologue  of  carbolic 
acid,  and  the  other  component  of  creosote,  is  not  known  to  possess  preservative 
properties. 

Creosoting,  or  Bethell's  process,  is  the  most  valuable  of  all  the  well-tried  pro- 
cesses in  this  case.  For  railway  sleepers  eight  pounds  of  oil  per  cubic  foot  of 
timber  is  sufficient.  *  If  the  timber  is  dry,  a  co'ating  of  coal-tar,  paint,  or  resin- 
ous substance,  is  valuable. 

A  Mr.  Heinmann,  of  New  York  City,  proposes  the  following  process,  which 
appears  to  be  very  promising  : — 

The  sap  is  first  expelled  and  then  the  timber  is  injected  with  common  rosin. 
The  latter  is  introduced  while  in  a  liquid  state,  under  high  pressure,  while  in 
vessels  especially  constructed  for  the  purpose. 

In  an  experiment  made  by  Prof.  Ogden,  one  cubic  foot  of  green  wood  ab- 
sorbed 8.96  pounds  of  rosin,  while  a  cubic  foot  of  well-seasoned  wood  absorbed 
only  2.66  pounds.  The  strength  of  the  timber  was  increased  by  this  process, 
as  is  shown  by  the  following  experiment : — 


Woo 

D  TREATED  WITH 

ROSIN. 

Wooi 

>  IN  ITS  NATURA] 

j  STATE. 

Breaking 

Breaking'!"' 

Weight 

Quality. 

Grain. 

Weight. 

Quality. 

Grain. 

Pounds. 

Pounds. 

163.5 

Checked. 

Straight. 

98.5 

Sound. 

Slant. 

193 

Sound. 

u 

103.0 

« 

171.5 

tt 

u 

116.0 

Straight. 

72.5 

Checked. 

Cross. 

57.5 

u 

57.5 

u 

Slant. 

46.00 

u 

57.5 

u 

M 

46.0 

It 

121.0 

(i 

U 

71.0 

(( 

155.5 

ft 

Cross. 

84.0 

(« 

It  is  found  by  experiment  that  wood  thus  treated  is  not  as  flammable  as  air- 
dried  wood.  This  is  accounted  for  from  the  fact  that  a  kind  of  inflammable 
slag  is  deposited  over  the  surface  immediately  after  the  rosin  begins  to  burn. 

The  chief  advantages  which  are  claimed  for  this  method  are  more  theoretical 
than  practical,  as  it  has  not  yet  had  sufficient  time  to  test  its  practical  merits, 
and  it  may,  like  many  other  processes,  disappoint  the  hopes  of  its  strongest  ad- 
vocates and  well-wishers. 

7.  TIMBER  CONSTANTLY  WET  IN  SALT  WATER.— We  have  not  to 
guard  against  decay  when  timber  is  in  this  situation.  Teredo  navalis,  a  mollugk 
of  the  family  Tubicolaria,  Lam  ,  soon  reduces  to  ruins  any  unprotected  sub- 
marine construction  of  common  woods.  We  quote  from  a  paper  read  before 
the  "Institute  of  Civil  Engineers,"  England,  illustrating  the  ravages  of  this 
animal : — 

"  The  sheeting  at  Southend  pier  extended  from  the  mud  to  eight  feet  above 
low-water  mark.  The  worm  destroyed  the  timber  from  two  feet  below  the 


*  Jour*    Frank.  /»«£.•,  voL  xliv.,  p.  275. 


238  APPENDIX. 

surface  of  the  mud  to  eight  feet  above  low-water  mark,  spring- tide ;  and  out  of 
38  fir-timber  piles  and  various  oak-timber  piles,  not  one  remained  perfect  after 
being  up  only  three  years."  *  Specimens  of  wood,  taken  from  a  vessel  that 
had  made  a  voyage  to  Africa,  are  in  the  museum,  and  show  how  this  rapid  de- 
struction is  effected. 

"  None  of  our  native  timbers  are  exempt  from  these  inroads.  Robert  Stephen- 
son,  at  Bell  Rock,  between  1814  and  1843, f  found  that  green  heart  oak,  beef- 
wood,  and  bullet-tree  were  not  perforated,  and  teak  but  slightly  so.  Later 
experiments  show  that  the  "  jarrah  "  of  the  East,  also,  is  not  attacked.:}:  The 
cost  of  these  woods  obliges  us  to  resort  to  artificial  protection. 

"  The  teredo  never  perforates  below  the  surf  ace  of  the  sea-bottom,  and  proba- 
bly does  little  injury  above  low- water  mark.  Its  minute  orifice,  bored  across 
the  grain  of  the  timber,  enlarges  inwards  to  the  size  of  the  finger,  and  soon  be- 
comes parallel  to  the  fibre.  The  smooth  circular  perforation  is  lined  through- 
out with  a  thin  shell,  which  is  sometimes  the  only  material  separating  the  ad- 
jacent cells.  The  borings  undoubtedly  constitute  the  animal's  food,  portions 
of  woody  fibre  having  been  found  in  its  body.  §  While  upon  the  surface  only 
the  projecting  siphuncles  indicate  the  presence  of  the  teredo,  the  wood  within 
may  be  absolutely  honey- combed  with  tubes  from  one  to  four  inches  in 
length. 

"It  was  naturally  supposed  that  poisoning  the  timber  would  poison  or  drive 
away  the  teredo,  but  Kyan's,  and  all  other  processes  employing  solutions  of 
the  salts  of  metals  of  alkaline  earths,  signally  failed.  This,  however,  is  not 
surprising.  The  constant  motion  of  sea-water  soon  dilutes  and  washes  away 
the  small  quantity  of  soluble  poison  with  which  the  wood  has  been  injected. 
If  any  albuminate  of  a  metallic  base  still  remains  in  the  wood,  the  poisonous 
properties  of  the  injection  have  been  destroyed  by  the  combination.  More- 
over, the  lower  vertebrates  are  unaffected  by  poisons  which  kill  the  mammals. 
Indeed,  it  is  now  known  that  certain  of  the  lower  forms  of  animal  life  live  and 
even  fatten  on  such  deadly  agents  as  arsenic.  || 

"  Coatings  of  paint  or  pitch  are  too  rapidly  worn  away  by  marine  action  to  be 
of  much  use,  but  timber,  thoroughly  creosoted  with  ten  pounds  of  dead  oil 
per  cubic  foot,  is  perfectly  protected  against  teredo  navalis.  All  recent  au- 
thorities agree  upon  this  point.  In  one  instance,  well  authenticated,  the  mol- 
lusk  reached  the  impregnated  heart-wood  by  a  hole  carelessly  made  through  the 
injected  exterior.  The  animal  pierced  the  heart-wood  in  several  directions, 
but  turned  aside  from  the  creosoted  zone.^f  The  process  and  cost  of  "  creo- 
soting"  have  already  been  discussed." 

A  second  destroyer  of  submarine  wooden  constructions  is  limnoria  terebrans, 
(or  L.  perforata,  Leach)  a  mollusk  of  the  family  Assellotes,  Leach,  resembling 
the  sow-bug.  It  pierces  the  .hardest  woods  with  cylindrical,  perfectly  smooth, 
winding  holes,  ^Vth  to  i^th  of  an  inch  in  diameter,  and^about  two  inches  deep.** 
From  ligneous  matter  having  been  found  in  its  viscera,  some  have  concluded 
that  the  limnora  feeds  on  the  wood,  but  since  other  mollusks  of  the  same  ge- 

*  Civ.  Eng..  Jour.,  vol.  xii.,  p.  382. 

t  The  Builder  for  1862,  p.  511. 

$  Civ.  Eng.  Jour.,  vol.  xx.,  p.  17. 

§  Civ.-  Eng.  Jour.,  vol.  xii.,  p.  382.    Also  Diet,  Univ.  d'Hist.  ''Natur.  tome  xii. 

||  British  and  Foreign  Medical  Review. 

*f  Civ.  Eng,  Jour.,  vol.  xii.,  p,  191. 

**  Dictt  Univ.  d'Hist.  Natur. 


APPENDIX.  239 

nus,  Pholas,  bore  and  destroy  stone -work,  the  perforation  may  serve  only  for 
the  animal's  dwelling.  The  lumnoria  seems  to  prefer  tender  woods,  but  the 
hardest  do  not  escape.  Green-heart  oak  is  the  only  known  wood  which  is  not 
speedily  destroyed.*  At  the  harbor  of  Lowestoft,  England,  square  fourteen  - 
inch  piles  were,  in  three  years,  eaten  down  to  four  inches  square. f 

While  all  agree  that  no  preparation,  if  we  except  dead  oil,  has  repelled  the 
limnoria,  an  eminent  engineer  has  cited  three  cases  in  which  that  agent  afford- 
ed no  protection.:}: 

We  do  not  find  that  timber  impregnated  with  water-glass  has  been  tested 
against  this  subtle  foe.  The  experiment  is  certainly  worthy  of  a  trial. 

A  mechanical  protection  is  found  in  thickly  studding  the  surface  of  the  tim- 
ber with  broad-headed  iron  nails.  This  method  has  proved  successful.  Oxy- 
dation  rapidly  fills  the  interstices  between  the  heads,  and  the  outside  of  the 
timber  becomes  coated  with  an  impenetrable  crust,  so  that  the  presence  of  the 
nails  is  hardly  necessary. 

In  conclusion,  we  cannot  but  express  surprise  that  so  little  is  known  in  this 
country  concerning  preservative  processes.  Their  employment  seems  to  excite 
very  little  interest,  and  the  very  few  works  where  they  are  being  tested  at- 
tract hardly  any  attention.  Those  railroads  which  have  suspended  their  use 
assign  no  reasons,  and  those  upon  which  the  timber  is  injected  publish  no  re- 
ports concerning  the  advantages  of  their  particular  methods.  Even  the  Na- 
tional Works,  upon  which  Kyan's  process  was  formerly  employed,  have  laid  it 
aside,  and  now  subject  lumber  to  dampness  and  alternations  of  wetness  and  dry- 
ness,  without  any  preparation  beyond  seasoning.  When  sleepers  cost  fifty 
cents  and  creosoting  thirty  cents  each,  it  is  cheaper  to  hire  money  at  seven  per 
cent. ,  compound  interest,  than  to  lay  new  sleepers  at  the  end  of  seven  years. 
Allowing  any  ordinary  price  for  the  removal  of  the  old  and  laying  down  the 
new  ties,  the  advantage  of  using  Bethell's  process  seems  evident.  If  some 
cheaper  method  will  produce  the  same  effects,  the  folly  of  neglecting  all  means 
which  aim  at  increasing  the  durability  of  the  material  is  still  more  palpable. 

*  Civ.  Eng,  Jour. ,  vol.  xxv.,  p.  206, 
t  Ibid.,  vol.  xvi.,  p.  76. 
%  Ibid.,  vol.  xxv.,  p.  206. 


240 


APPENDIX. 


8.  THE  FOLLOWING  IS  A  SUMMARY  of  the  different  processes  that  have 
been  invented  from  time  to  time,  with  the  names  of  their  inventors :  — 


Names  of  Inventors. 

Chemicals  Used. 

Manner  of  using  them. 

Bethell 

Creosote   or  pitch-oil 

By  injection. 

1  1                    1  1 

it             tt 
It               tt 
(t               tt 
u             tt 
tt             (( 

1  1               tt 

tt             tt 

tt             t« 

tt             tt 

"     boHing. 

tt           tt 

«t           tt 

Cf                   tt 

By  means  of  a  vacuum  and 
injection. 
By  immersion,  and  by  fire 
or  burning. 
By  saturation. 
External  application. 

Kyan 

Chloride  of  mercury 

Margery  

Sulphate  of  copper 

Chloride  of  zinc  

Liquid  silicate  of  potassa  

LeGras 

Manganese,  lime,  and  creosote  .  . 
Solution  of  acetate  of  copper.  .  .  . 
Sulphate  of  iron,  carbonate  of  } 

Margary  

Payne  •] 

Bouchere.           •) 

Pyrolignate  of  iron,    sulphate  j 
of  copper                                    i 

Gemini 

Tar              .  .                            

Heinmann  

Rosin,  or  colophony  

Earle  

Sulphate  of  copper  

Tregold 

Sulphate  of  iron 

S.  Beer  

Borax  

Dorset  and  Blythe 

Huting     and     j 
Boutigny  "j 

Same  as  Bouchere  .  •} 

Oil   of  schist,    tar,    pitch,  and  \ 
shellac  ) 

Vernet  

Arsenic  

Salt      .               

APPENDIX    II. 

TABLE 

Of  the  Mechanical  Properties  of  the  Materials  of  Construction. 


NOTE. — The  capitals  affixed  to  the  numbers  in  this  table  refer  to  the  following  authorities  : — 


B.  Barlow.    Report  of  the  Commissioners  of  j 
the  Navy,  etc. 

Be.  Bevan. 

Bn.  Buchanan. 

Br.  Belidor,  Arch.  Hydr. 

Bru.  Brunei. 

C.  Couch.  . 
01.  Clark. 

D.  Darcel,  Annales  for  1858. 

D.  W.    Daniell  and  Wheatstone.    Report  on 
the  stone  for  the  Houses  of  Parliament. 

E.  Bads. 

F.  Fairbairn. 
G-.  Grant. 

H.  Hodgkinson.  Report  to  the  British  Asso- 
ciation of  Science,  etc. 

Ha.  Haswell.  Eng.  and  Mech.  Pocket-Book, 
1869. 

J.  Journal  of  Franklin  Institute,  vol.  XIX., 
p.  451. 

K.  Kirwan. 


Ki.  Kirkeldy. 
La.  Lame. 

M.  Mischembroeck.    Introd.  ad.  Phil.  Nat  I 
Ma.  Mallet. 
ML  Mitis. 
Mt.  Mushet. 
Pa.  Colonel  Pasley. 
R.  Roudelet.     L'Art  de  Batir,  IV. 
Ro.  Roebling. 

Re.  Reunie.     Phila.  Trans.,  etc. 
S.  Styffe.     On  Iron  and  Steel. 
T.  Thompson. 
Te.  Telford. 

Tr.  Tredgold.    Essay  on  the  Strength  of  Cast 
Iron. 

W.  Watson. 
Wa.  Major  Wade. 
Wn.  Wilkinson. 


*  Calculated  from  the  experiments  of  Fair- 
bairn  and  HodgkinBon. 


U 

g1 

11 

*o 

*fl 

ft£  . 

h. 

"S 

NAMES  OF  MATERIALS. 

£     E^ 

c  ^     ^ 

jo  o  p3 

QJ   -4^  W 

&    6W 

'o-S 

•—    Q 

13 

.SP-2 

1  s\g 

1* 

rt 

11 

ill 

IS 

II 

METALS. 

Antimony  — 
Cast 

281-25 

1,066  M. 

Bismuth            

613-87 

3,250  M. 

Brass  — 

Cast                         .  . 

525  '00 

17968  Re 

10,304  Re. 

9,170,000 

Wire-drawn 

534-00 

14,230,000 

Copper  — 

Cast              

537-93 

19,072 

29,272  Re. 

Sheet 

549-06 

32,184 

560*00 

61  228 

In  Bolts                .   .  . 

48,000 

Iron. 

Cast  Iron. 

48,240  T. 

18,014,400  T. 

Carron,  No.  2  — 
Cold  Blast  

441-62 

16,683  H. 

106,375  H. 

38.556  H. 

17,270,500  H.* 

Hot  Blast  

440-37 

13,505  H. 

108,540  H. 

37,503  H. 

16,085,000  H.* 

Carron,  No.  3  — 
Cold  Blast  

443-37 

14,200  H. 

115,442  H. 

35,980  F.* 

16,246,966  F. 

Hot  Blast..., 

441-00 

•  17,775  H. 

133,440  H. 

42,687  F.* 

17,873,100  F. 

16 


242 


APPENDIX 
TABLE —  Continued. 


NAMES  OF  MATERIALS. 

Weight  of  one 
cubic  ft.  in  Ibs. 
8 

Tenacity  per  sq. 
inch  in  Ibs. 
T. 

Crushing  Force 
per  square  inch 
in  Ibs. 
C. 

Modulus  of 
Rupture. 
R. 

Coefficient  of 
Elasticity. 
E. 

Iron. 
Cast  Iron. 

Devon,  No.  3— 
Cold  Blast                   .... 

455  •  93 

36,288  H  * 

22  907  700  H 

Hot  Blast  

451  81 

29  107  H. 

145,435  H. 

43,497  H.* 

22,473,650  H. 

Buffery,  No.  1— 
Cold  Blast  

Hot  Blast 

442-43 
437'37 

17,466  H. 
13  434  H 

93,366  H. 
86397  H 

37,503  H.* 
35,316  H  * 

15,381,200  H. 
13  730  500  H. 

Coed  Talon,  No.  2— 
Cold  Blast 

434-06 

18  855  H 

81  770  H 

33,453  F.* 

14,313  500  F. 

Hot  Blast  

435-50 

16,676  H 

82,739  H. 

33,696  H.* 

14,322,500  F. 

Elsicar,  No.  1— 
Cold  Blast  

439-37 

34,587  F.* 

13,981,000  F. 

Milton,  No.  1— 
Hot  Blast  

436-00 

29,889  F.* 

11,974,500  F. 

Muirkirk,  No.  1— 
Cold  Blast  

444-56 

36,693  F.* 

14,003,550  F. 

Hot  Blast 

434  "56 

33,850  P.* 

13,294,490  F. 

Morris  Stirling's  2d  quality.  . 

25,764 

119,000 

(See  also  p.  52.) 
Gun  Metal- 
American.   

1 

14,000 
to 

I- 

27,548,000  Wa. 

Extra  Specimens  

595-00 

1 

34,000  Wa. 
45  970  Wa. 

} 

Steel. 
Hammered  Cast  Steel,  from 

91,000  )  o 

31,359,000  S. 

Tempered  ... 

171  000  S 

Bessemer  Steel,  from  Hogbo, 
marked  10  . 

r!40  945  S 

31,819,000  S. 

Bessemer  Steel,  Eng.  Mean  of 
four  experiments    

485-37 

88,415  F. 

225,568  F. 

29,215,000  F. 

Naylor,  Tickers  &  Co.  Cruci- 
ble Steel  

488-70 

108  099  F 

225,568  F. 

30,278,000  F. 

Mushet's  Steel- 
Soft 

492-50 

93  616  F 

31,901,000  F. 

Cast  Steel- 
Soft 

486-25 

120  000 

Not  Hardened  

198,944  Wa. 

Mean  Temper 

381,985  Wa 

490'GO 

150  000 

29,000,000 

Steel  Wire  Rope- 
Fine  Wire  

40,000  Ro. 

Cast  Steel    .                .   . 

242  100 

Wrought  Iron. 
English  

481-20 

See  page  25. 
57  300  La 

) 

•\              O 

In  Bars                                    \ 

475-50 

57  300  La 

>  See  page  53. 

Hammered  

487-00 

67  200  Bra 

1 

®® 

60  480  La 

,       §g 

Swedish,  in  bars  

71  680  R 

'         cfxT 

English,   in    wire    1-10    inch 
diarn  

\ 

80,000  Te. 
96  000  Te 

<K<r< 

Russian,  in  wire;  diam.  1-2U 
to  1-30  inch  

{ 

134,000  La. 
203,000  La. 

J          H 

APPENDIX. 
TABLE—  Continued. 


243 


NAMES  OF  MATERIAL. 

Weight  of  one 
cubic  ft.  in  Ibs. 
S 

g1 

Su 

•S-S 

I! 

Crushing  Force 
per  square  inch 
in  Ibs. 
C. 

gg'tf 

IS 

Coefficient  of 
Elasticity. 
E. 

Wrought  Iron. 

Rolled  in  sheets  and  cut  cross- 
wise                        

40  320  Mi 

31  360  Mi 

In  chains,  oval  links,  iron  % 

48  IfiO  Br 

§ 

Wire  American  

73'  600  Ha 

0- 

Lake      Superior     and     Iron 
Mountain  Charcoal  Bloom. 

90  000  Ha. 

I 

47  909  J 

52  099  J 

Salisbury   Ct     40  exp  

58'009  J* 

,     s 

58  400  J 

5" 

Phillipsburgh  Wire,  Pa. 
(  0-333  in.,  13  exp... 

84,186  J. 

of 

Diam.-{  0.190  in.,  5  exp.... 
\  0.156  in.,  5  exp  
Mean  of  188  rolled  bars 



73,888  J. 
89,162  J. 
57  557  Ki 

1 

Mean   of   167  plates  length- 

50  737  Ki 

Mean  of  160  plates  crosswise  . 

46  171  Ki 

60  364  Ki 

• 

Swedish    forged  

41,000  Ki. 

Hammered    Bessemer    Iron, 

50,000  Ki. 

32  320  000  S 

Low    Moor    Rolled    Puddled 

31  976  000  S 

Rolled  Iron,    Swedish,   char- 
coal heath  

65,000  S. 

27  000  000  S 

717  -45 

1  824  Re 

Lead  Wire                

705  '12 

2,581  M 

Silver  standard.         

644-50 

40,902  M. 

455-68 

5  322  M 

4  608  000  Tr 

2inc                                

439-25 

13  680  000  Tr 

STONE  —  NATtrKAL  AND  ABTI- 

FICIAL. 

Granites. 

164 

10,914  Re. 

166 

6,356  Re 

Killincv  very  felspathic.  .  .    . 

10,780  Wn. 

16(5 

12,286  F. 

Sandstones. 

6,493  Bn. 

158 

6,630  Re. 

Derby    Grit,    a    red,    friable 

148 

3,142  Re. 

156 

4,345  Re. 

Limestones. 
Limestone,  Magnesian  (Graf- 

[ 

17,000  E. 

'  ] 

Same  as  W^t. 

Iron       V 

ton,  111.)  

) 

( 

244 


APPENDIX. 
TABLE —  Continued. 


NAMES  OF  MATERIALS. 

Weight  of  one 
cubic  ft.  in  Ibs. 
8 

£ 

£«•  . 

IT 

§1 

Crushing  Force 
per  square  inch 
in  Ibs. 
C. 

Modulus  of 
Rupture. 
R. 

Coefficient  of 
Elasticity. 
E. 

Limestones. 

Limestone,  compact  
Limestone,    Kerry,    Listowel 
Quarry,  Eng  .  .  . 

162 

7,713  Re. 

18,043  Wn. 
501  Re. 

3,216  Re. 
9,681  Re. 
9,219  Re. 
3,792  Re. 

1.062 
2,664 

i'/? 

ifi 

j 

11,202  B. 

j-  12,156  B. 

20,886  B. 
9,336  B. 

25,200,000  T. 

ffr^iil 

rAbraf 

®*  California 

1,152,000  B. 

1,644,800  B. 

2,  601,600  B. 
1,353,600  B. 

Chalk  

Other  Stones. 

Alabaster  (Oriental),  white  .  . 
Marble,  statuary       .... 

170 

Do.     white  Italian,  veined. 
Do.      black  Galloway.   . 

165 
168 
151 

Portland  Stone  (Oolite)  
Valentia,  Kerry  (slate  stone).. 
Green    Stone,    from    Giant's 
Causeway  

10,943  Wn. 
17,220  Wn. 
25,500  Ma. 
14,000  Ma. 
2,010  Re. 

808  Re. 
562  Re. 
j     800  to 
1  4,000  Ha. 
1,717  Re. 
2,177  Ha. 

521  01. 

J     500  to 
\     800  Ha. 

1 

J  1,000  to 
1  5,900  G. 

334  Re. 

Quartz  Rock,Holyhead(across 
lamination)  

Quartz  Rock  (parallel  to  lami- 
nation)   

Gravel 

120 
158 

135-5 
130-31 

.-•• 

Green  Moor.        .... 

Artificial  Stone. 
Brick    red  

280 
300 

Brick,  pale  red  ...             .... 

Brick,  common  

Bire  Brick,  Stourbridge  

Brick,  Stock  

Bricks  set  in  cement  (bricks 
not  very  hard)  .... 

Brick  Masonry,  common  

Cement,  Portland,  with  sand. 
Cement,   Portland,    with  no 

j    92  to 

1  284  D. 

f  427  to 
1711 

Cement,  Portland  

Chalk  

116-81 
153-31 

107 

47-37 
50-00 
49-56 
43-12 
55-81 
51-37 
67-51 
53-37 
45-12 

Glass  Plate 

9,420 
50 

16,000  Be. 
14,186  M. 
19,500  Be. 

j-  17,  207  B. 
12,396 

Mortar  

j     120  to 
j     240  Ha. 

"  "6,859  H'." 

j     8,6&3  H. 
1      9,363  H. 
7,158  H. 

TIMBER. 
Acacia,  English  

Alder                  .... 

Apple  Tree  

•L   (  Ordinary  state 

Asbl  Very  dry  

Bay  Tree          .      .         .... 

(  Ordinary.  ...        .... 

15,784  B. 
17,850  B. 

7.733  H.  | 
9,363  H.  f 

B9ecMvervdrv... 

APPENDIX. 
TABLE—  Continued. 


245 


NAMES  OF  MATERIALS. 

Weight  of  one 
cubic  ft.  in  Ibs. 
S 

£ 

Ij. 

£>~H 

If 
II 

Crushing  Force 
per  square  inch 
in  Ibs. 
C. 

Modulus  of 
Rupture. 
R. 

Coefficient  of 
Elasticity. 
E. 

TIMBEE. 

49-50 

15  000 

j  4,533  H.  1 

10  920  B 

1  *S62  400  "R 

Birch,  American  

40-50 

1  6,402  H.  | 
11,663  H. 

9,624  B. 

1  257  600  B 

Box  dry 

6Q-00 

19  891  B 

10  299  H 

Bullet  Tree  (Berbice)  

64  "31 

15636  B 

2  610  600  B 

Cane                             

25-00 

6  300  Be 

Cedar,  Canadian  

56'  81 

11  400  Be 

5,674  H. 

Crab  Tree  

47'80 

6,499  H. 

Deal- 
Christiana  Middle  

43-62 

12400 

9864  B 

1  672  000  B 

Norway  Spruce  

21-25 

17,600 

English  

29-37 

7000 

Red  

5,748  H. 

White  

6,741  H. 

Elder                

43-43 

10  230 

8467  H 

35-75 

13  489  M 

10  331  H 

6  078  B 

699  840  B 

Fir- 
New  England  

34'56 

6,612  B. 

2  191  200  B 

Riga         

47-06 

j  11,549  to 

5,748  to 

6,648  B. 

1,328,800 

Hazel  .. 

53-75 

|  12,857  B. 
18  000  Be 

6,586  H. 

7,572  B. 

869,600  B. 

Lance  Wood 

63'87 

24  696 

Larch- 
Green  

32-62 

10  220  B 

3  201  H. 

4,992  B 

897  600  B 

Dry  

35-00 

8,900  B. 

5,568  H. 

6,894  B. 

1,052  800  B 

Lignum-vitae  .... 

76-25 

11  800  M 

Mahogany,  Spanish  
Maple,  Norway  

50-00 
49-56 

16,500 
10584 

8,198  H. 

English  

58-37 

17  300  M. 

J  4,684  to 

10,032  B. 

1  451  200  B 

Canadian  

Dantzio 

54-50 
47-24 

10,253 
12  780 

1  9,509  H. 
j  4.231  to 
\  9,509  H. 

10,596  B. 
8742  B. 

2,148,800  B. 
1  191  200  B 

Adriatic  

62-06 

8,298  B. 

974  400  B 

African  Middle. 

60-75 

13  566  B. 

2283200  B 

Pear  Tree 

41-31 

7518H 

Pine- 
Pitch 

41-25 

7818  M 

9792 

1  225  600  B 

Red  

41-06 

5,375  H. 

8,946  B. 

1,840,000  B. 

American  Yellow 

28-81 

5,445-H. 

1,600  000  Tr 

Plum.Tree  

49-06 

11,351 

j  3,657  to 

Poplar 

23-93 

7200 

1  9,367.H. 
j  3,107  to 

Teak,  dry. 

41-06 

15  000  B. 

|  5,124  H. 
12.101  H. 

14,722  B. 

2,414,400  B 

Walnut 

41  •  93 

8  130  M 

6635H. 

306000 

Willow,  dry  

24-37 

14,000  Be. 

ERRATA. 


Page  14,  line  9,  for  P  =  13,  934,000y  =  2,907,432,000^,  read, 

P  =  13,934,000y  -  2,907,432,000^' 

"     22,     "    5,  for  ffit,  read,  d&. 

"      "    at  the  middle  of  the  page,  for  ResiUance  of  Prisms,  read,  Rezi- 
fance  of  Prisms. 


"  "  10  from  the  bottom,  for  t»,  read,        & 

6  6 

26,  u    3,  for  exponetials,  read,  exponentials. 

36,  "  10,  for  rods  a?1  rivet  iron,  read,  rods  of  rivet  iron. 

53,  "2,  for  No.  1,  read,  No.  2. 

56,  "3  from  the  bottom,  for  z,  read,#. 

68,  "       at  the  bottom  of  the  table,  for  Mean  6852,  read,  685  '2. 

72,  "  15,  for  equation  (26),  read,  equation  (24).  ' 

96,  "    1,  for  82,  read,  86. 


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